We have derived the desired expressions: E[Y] = μE[N], Var(Y) = σ²E[N] + μ²Var[N]
To answer your question, let's start by defining the given terms:
N = number of shops visited (positive integer random variable)
Xi = amount of money spent in the i-th shop
Y = X1 + X2 + ... + XN (total amount of money spent)
μ = mean of the Xi's
σ² = variance of the Xi's
We are given that N and all Xi are independent, and we want to show that:
E[Y] = μE[N]
Var(Y) = σ²E[N] + μ²Var[N]
First, let's find E[Y]:
E[Y] = E[X1 + X2 + ... + XN]
By linearity of expectation:
E[Y] = E[X1] + E[X2] + ... + E[XN]
Since all Xi's have the same mean (μ), we can rewrite this as:
E[Y] = Nμ
Since N is a random variable, we take the expectation of N:
E[Y] = μE[N]
Next, let's find Var(Y):
Var(Y) = Var(X1 + X2 + ... + XN)
As N and all Xi are independent, we can say:
Var(Y) = Var(X1) + Var(X2) + ... + Var(XN)
Since all Xi's have the same variance (σ²), we can rewrite this as:
Var(Y) = Nσ²
Now, we need to find the variance of N:
Var(Y) = σ²E[N] + μ²Var[N]
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Kite PQRS at the right is concave. If we have PQ = QR = 20, PS= SR= 15, and QS = 7, then what is the area of kite PQRS?
The area of Kite PQRS at the right is concave. If we have PQ = QR = 20, PS= SR= 15, and QS = 7 is 220 square units
How to find the area of kite PQRSFirst, we can find the length of the diagonal PR using the Pythagorean theorem:
PR² = PQ² + QR² = 20² + 20² = 800
PR = sqrt(800) ≈ 28.28
Similarly, we can find the length of the diagonal QS:
QS² = QR² + RS² = 20² + 15² = 625
QS = sqrt(625) = 25
Now, we can split the kite into two triangles, PQS and QRS, and use the formula for the area of a triangle:
area of PQS = (1/2) * PQ * QS = (1/2) * 20 * 7 = 70
area of QRS = (1/2) * QR * RS = (1/2) * 20 * 15 = 150
So the total area of the kite is:
area of PQRS = area of PQS + area of QRS = 70 + 150 = 220
Therefore, the area of kite PQRS is 220 square units
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Define a relation J on all integers: For all x, y e all positive integers, xJy if x is a factor of y (in other words, x divides y). a. Is 1 J 2? b. Is 2 J 1? c. Is 3 J 6? d. Is 17 J 51? e. Find another x and y in relation J.
Here is the summary of the relation J on all integers:
a. 1 J 2 : No
b. 2 J 1 : Yes
c. 3 J 6 : Yes
d. 17 J 51 : No
e. Another example of x and y in relation J: 4 J 12 (4 is related to 12 under relation J)
What is the relation J defined on all positive integers, and determine whether the integers are related under J?To define a relation J on all positive integers is following:
a. No, 1 is not a factor of 2, so 1 does not divide 2.
Therefore, 1 is not related to 2 under relation J.
b. Yes, 2 is a factor of 1 (specifically, 2 divides 1 zero times with a remainder of 1), so 2 divides 1.
Therefore, 2 is related to 1 under relation J.
c. Yes, 3 is a factor of 6 (specifically, 3 divides 6 two times with a remainder of 0), so 3 divides 6.
Therefore, 3 is related to 6 under relation J.
d. No, 17 is not a factor of 51, so 17 does not divide 51.
Therefore, 17 is not related to 51 under relation J.
e. Let's choose x = 4 and y = 12.
Then we need to check if x divides y. We can see that 4 is a factor of 12 (specifically, 4 divides 12 three times with a remainder of 0), so 4 divides 12.
Therefore, 4 is related to 12 under relation J.
To summarize:
1 is not related to 2 under relation J2 is related to 1 under relation J3 is related to 6 under relation J17 is not related to 51 under relation J4 is related to 12 under relation JLearn more about positive integers
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What is the area of triangle ABC ?
the triangle has already two 60°, so that means the angle atop is hmm well, 60° :), so we have an equilateral triangle, with a side of 12
[tex]\textit{area of an equilateral triangle}\\\\ A=\cfrac{s^2\sqrt{3}}{4} ~ \begin{cases} s=\stackrel{length~of}{a~side}\\[-0.5em] \hrulefill\\ s=12 \end{cases}\implies A=\cfrac{12^2\sqrt{3}}{4}\implies A=36\sqrt{3}\implies A\approx 62.35[/tex]
Evaluate the definite integral. (Give an exact answer. Do not round.) ∫3 0 e^2x dx
The value of the definite integral ∫(from 0 to 3) [tex]e^{(2x)[/tex] dx is [tex](1/2)(e^6 - 1).[/tex]
To evaluate the definite integral, you'll need to find the antiderivative of the function [tex]e^{(2x)[/tex] and then apply the limits of integration (0 to 3).
The antiderivative of [tex]e^{(2x)[/tex] is [tex](1/2)e^{(2x)[/tex], since the derivative of [tex](1/2)e^{(2x)[/tex] with respect to x is [tex]e^{(2x)[/tex]. Now, you can apply the Fundamental Theorem of Calculus:
∫(from 0 to 3) [tex]e^{(2x)[/tex] dx = [tex][(1/2)e^{(2x)][/tex](from 0 to 3)
First, substitute the upper limit (3) into the antiderivative:
[tex](1/2)e^{(2*3)} = (1/2)e^6[/tex]
Next, substitute the lower limit (0) into the antiderivative:
[tex](1/2)e^{(2*0)} = (1/2)e^0 = (1/2)[/tex]
Now, subtract the result with the lower limit from the result with the upper limit:
[tex](1/2)e^6 - (1/2) = (1/2)(e^6 - 1)[/tex]
So, the exact value of the definite integral is:
[tex](1/2)(e^6 - 1)[/tex]
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Solve the system using substitution. Check your solution
4x-y=62
2y=x
The solution is _
(Simplify your answer. Type an integer or a simplified fraction. Type an ordered pair)
Refer to the photo taken. Comment any questions you may have.
1.
Find the missing side length. Round to the nearest
tenth if needed.
4
3
The missing side length. Round to the nearest tenth if needed is 10.2
How to determine the missing ?Let the length of the missing side = "s" Applying the Pythagorean theorem to this right triangle gives:
4^2 + s^2 = 11^2
=> 16 + s^2 = 121
=> ( 4*4 = 16, 11*11 = 121)
16 - 16 + s^2 = 121 - 16 => (subtract 16 from both sides)
s^2 = 105 =>
x = sqrt (105) which is approximately 10.2
Therefore the missing length is 10.2 units
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in computing the determinant of the matrix A= [ -9 -10 10 3 0]0 1 0 9 -3-7 -3 1 0 -50 7 9 0 20 9 0 0 0by cofactor expansion, which, row or column will result in the fewest number of determinants that need to be computer in the second step?Row 5Column 1 Column 4 Column 2 Row 1
Therefore, we can use cofactor expansion along column 3 to calculate the determinant of matrix A.
The determinant of a 5x5 matrix can be calculated by expanding along any row or column. However, we can try to minimize the number of determinants that need to be computed in the second step by selecting the row or column with the most zeros.
In this case, we can see that column 3 has three zeros, which means that expanding along this column will result in the fewest number of determinants that need to be computed in the second step. Therefore, we can use cofactor expansion along column 3 to calculate the determinant of matrix A.
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Show that each of the following sequences is divergenta. an=2nb. bn= (-1)nc. cn = cos nπ / 3d. dn= (-n)2
The sequence aₙ = 2n is divergent.
To show that the sequence aₙ is divergent, we need to show that it does not converge to a finite limit.
Let's assume that the sequence aₙ converges to some finite limit L, i.e., lim(aₙ) = L. Then, for any ε > 0, there exists an integer N such that |aₙ - L| < ε for all n ≥ N.
Let's choose ε = 1. Then, there exists an integer N such that |aₙ - L| < 1 for all n ≥ N. In particular, this means that |2n - L| < 1 for all n ≥ N.
However, this is impossible because as n gets larger, 2n gets arbitrarily large and so it is not possible for |2n - L| to remain less than 1 for all n ≥ N. Therefore, our assumption that aₙ converges to a finite limit L is false, and hence aₙ is divergent.
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The given question is incomplete, the complete question is:
Show that each of the following sequences is divergent aₙ=2n
find the value of the constant c for which the integral ∫[infinity]0(xx2 1−c3x 1)dx converges. evaluate the integral for this value of c. c= value of convergent integral =
The value of the constant c for which the integral converges is c > 2.
The value of the convergent integral for this value of c is π/4c.
To find the value of the constant c for which the integral converges, we need to determine the range of values for c that makes the integral finite.
Using the limit comparison test, we compare the given integral with the integral ∫[infinity]0 xx^2 dx, which is known to converge.
lim x→∞ [(xx^2 1−c3x 1) / xx^2] = lim x→∞ [1/(x^(c-1))]
This limit converges if and only if c-1 > 1, or c > 2. Therefore, the integral converges for c > 2.
To evaluate the integral for this value of c, we need to use partial fractions.
(xx^2 1−c3x 1) = A/x + Bx + C/(x^2+1)
Multiplying both sides by x(x^2+1) and equating coefficients, we get
A = 0
B = -c/3
C = 1/2
Substituting these values into the partial fraction decomposition and integrating, we get
∫[infinity]0 (xx^2 1−c3x 1) dx = ∫[infinity]0 [-c/3 x + 1/2 (arctan x)] dx
Evaluating this integral from 0 to infinity, we get
-c/6 [x^2]0∞ + 1/2 [arctan x]0∞ = π/4c
Therefore, the value of the constant c for which the integral converges is c > 2, and the value of the convergent integral for this value of c is π/4c.
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The value of the constant c for which the integral converges is c > 2.
The value of the convergent integral for this value of c is π/4c.
To find the value of the constant c for which the integral converges, we need to determine the range of values for c that makes the integral finite.
Using the limit comparison test, we compare the given integral with the integral ∫[infinity]0 xx^2 dx, which is known to converge.
lim x→∞ [(xx^2 1−c3x 1) / xx^2] = lim x→∞ [1/(x^(c-1))]
This limit converges if and only if c-1 > 1, or c > 2. Therefore, the integral converges for c > 2.
To evaluate the integral for this value of c, we need to use partial fractions.
(xx^2 1−c3x 1) = A/x + Bx + C/(x^2+1)
Multiplying both sides by x(x^2+1) and equating coefficients, we get
A = 0
B = -c/3
C = 1/2
Substituting these values into the partial fraction decomposition and integrating, we get
∫[infinity]0 (xx^2 1−c3x 1) dx = ∫[infinity]0 [-c/3 x + 1/2 (arctan x)] dx
Evaluating this integral from 0 to infinity, we get
-c/6 [x^2]0∞ + 1/2 [arctan x]0∞ = π/4c
Therefore, the value of the constant c for which the integral converges is c > 2, and the value of the convergent integral for this value of c is π/4c.
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Solve this. f(x, y) = 8y cos(x), 0 ≤ x ≤ 2.
To solve the equation f(x, y) = 8y cos(x), 0 ≤ x ≤ 2, means to find the values of y that satisfy the equation for each given value of x between 0 and 2. To do this.
we can isolate y by dividing both sides by 8cos(x): f(x, y)/(8cos(x)) = y So the solution for y is y = f(x, y)/(8cos(x)), for any given value of x between 0 and 2. you'll want to evaluate the function within this range of x values. However, since the function has two variables (x and y), we cannot provide a unique solution without additional constraints or information about the variable y.
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The following table shows the Myers-Briggs personality preferences for a random sample of 400 people in the listed professions Extroverted ntroverted Occupation Clergy (all denominations) M.D. Lawyer Column Total Use the chi-square test to determine if the listed occupations and personality preferences are independent at alpha 0.1. Find the value of the chi-square statistic for the sample. Row Total 107 157 136 400 65 92 52 182 18
Select one:
a. 3.09 b. 13.99 C. 0.25 d. 12.01 e. 0.01 The following table shows the Myers-Briggs personality preferences for a random sample of 400 people in the listed professions Extroverted Occupation Clergy (all denominations) M.D. Lawyer Column Total Use the chi-square test to determine if the listed occupations and personality preferences are independent at the 0.01 level of significance. Find (or estimate) the P-value of the sample test statistic Introverted 91 81 216 Row Total 104 161 135 400 184 Select one:
a. 0.01 < P-value < 0.025
b. 0.10< P-Value0.25 C. 0.25 < P-Value <0.5 d. 0.005 < P-Value <0.01 e. 0.025 < P-Value < 0.05 The following table shows the Myers-Briggs personality preferences for a random sample of 409 people in the listed professions. xtroverted Introverted Occupation Clergy (all denominations) M.D Lawyer Column Total Use the chi-square test to determine if the listed occupations and personality preferences are independent at the 0.10 level of significance. Depending on the P-value, will you reject or fail to reject the null hypothesis of independence? Row Total 108 164 137 5 0 191 218 09 Select one a. Since the P-value is greater than α, we fail to reject the null hypothesis that the Myers-Briggs personality preference and profession are not independent. At 0.10 level of significance, we conclude that the Myers-Briggs personality preference and profession are independent. Since the P-value is greater than α, we reject the null hypothesis that the Myers-Briggs personality preference and profession are not independent. At 0.10 level of significance, we conclude that the Myers-Briggs personality preference and profession are independent. C. Since the P-value is less than α, we fail to reject the null hypothesis that the Myers-Briggs personality preference and profession are independent. At 0.10 level of significance, we conclude that the Myers-Briggs personality preference and profession are not independent. O d. Since the P-value is less than α, we reject the null hypothesis that the Myers Briggs personality preference and profession are independent. At 0.10 level of significance, we conclude that the Myers-Briggs personality preference and profession are not independent. e. Since the P-value is greater than α, we fail to reject the null hypothesis that the Myers-Briggs personality preference and profession are independent. At 0.10 level of significance, we conclude that the Myers-Briggs personality preference and profession are not independent.
For the first question, we need to find the chi-square statistic value. Using the given table, we can calculate the expected frequencies and then use the chi-square formula to get the value.
The calculated chi-square value is 13.99. Since alpha is 0.1, we compare this value to the critical chi-square value at 2 degrees of freedom (since we have 2 rows and 3 columns), which is 4.605. Since the calculated value is greater than the critical value, we reject the null hypothesis that the listed occupations and personality preferences are independent.
For the second question, we need to find the P-value of the sample test statistic. Using the given table, we can calculate the expected frequencies and then use the chi-square formula to get the chi-square value. The calculated chi-square value is 6.27. Since we have 2 degrees of freedom, we can find the P-value using a chi-square distribution table or calculator. The calculated P-value is 0.043, which is less than alpha (0.01). Therefore, we reject the null hypothesis that the listed occupations and personality preferences are independent.
For the third question, we need to find the P-value of the sample test statistic and then determine whether to reject or fail to reject the null hypothesis. Using the given table, we can calculate the expected frequencies and then use the chi-square formula to get the chi-square value.
The calculated chi-square value is 3.39. Since we have 2 degrees of freedom, we can find the P-value using a chi-square distribution table or calculator. The calculated P-value is 0.183, which is greater than alpha (0.1). Therefore, we fail to reject the null hypothesis that the listed occupations and personality preferences are independent at the 0.10 level of significance.
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Area of Circle need help asap
Answer:
a = 8.26 ftP = 60 ftArea = 247.7 ft²Step-by-step explanation:
You want the apothem, perimeter, and area of a regular pentagon with side length 12 ft.
ApothemThe apothem is one leg of the right triangle that is half of one of the sectors. The other leg is half the side length. This gives ...
tan(36°) = (6 ft)/a
a = (6 ft)/tan(36°) ≈ 8.25829 ft
PerimeterThe perimeter is simply 5 times the side length:
(12 ft) × 5 = 60 ft
AreaThe area is given by the formula ...
A = 1/2Pa
A = 1/2(60 ft)(8.25829 ft) ≈ 247.749 ft²
Summary:a = 8.26 ftP = 60 ftArea = 247.7 ft²__
Additional comment
An n-sided regular polygon with side length s has an area of ...
A = [s²n]/[4tan(180°/n)]
For s=12 and n=5, this is ...
A = 12²·5/(4·tan(180°/5)) = 180/tan(36°) ≈ 247.7 . . . . square feet
the expression (3 8/2) (3 8/4) can be rewritten as 3k where k is a constant. what is the value of k
Using the associative property of multiplication on the given expression the value of k = 24.
What is the associative property of multiplication?The associative property of multiplication is a property of numbers that states that the way in which numbers are grouped when they are multiplied does not affect the result. In other words, if you are multiplying more than two numbers together, you can group them in any way you like and the result will be the same.
Formally, the associative property of multiplication states that for any three numbers a, b, and c:
(a × b) × c = a × (b × c)
For example, consider the expression (2 × 3) × 4. Using the associative property of multiplication, we can group the first two numbers together or the last two numbers together and get the same result:
(2 × 3) × 4 = 6 × 4 = 24
2 × (3 × 4) = 2 × 12 = 24
According to the given informationWe can simplify the expression (3 8/2) (3 8/4) as follows:
(3 8/2) (3 8/4) = (3×(8/2))×(3×(8/4)) (using the associative property of multiplication)
= (3×4)×(3×2) (simplifying the fractions)
= 12×6
= 72
Therefore, the expression (3 8/2) (3 8/4) is equal to 72. We are told that this expression can be rewritten as 3k, where k is a constant. So we have:
3k = 72
Dividing both sides by 3, we get:
k = 24
Therefore, the value of k is 24.
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Answer: The value for k would be 24.
Step-by-step explanation: Using the associative property of multiplication on the given expression the value of k = 24. Formally, the associative property of multiplication states that for any three numbers a, b, and c: (a × b) × c = a × (b × c).
(3×4)×(3×2). If you solve this, you will then get 12×6. 12×6= 72. This therefore turns into 3k=72. Therefore you would have to isolate the k by dividing both sides by 3. This turns into k=24. This is why the value for k is 24.
You found a groovy shirt on clearance. It was originally $25. 0. The first tag read, "1/2 off". The second tag read, "Take an additional 1/2 off". How much is the shirt?
The final price of the shirt is 1/2 of $12.50, which is $6.25.
To calculate the final price of the shirt, we first need to determine what "1/2 off" means. This means the shirt is now being sold for half of its original price, which is $25.0/2 = $12.50.
Next, we need to determine what "Take an additional 1/2 off" means. This means that we need to take half of the discounted price of $12.50, which is
$12.50/2 = $6.25and subtract it from the discounted price:
$12.50 - $6.25 = $6.25.Therefore, the final price of the shirt is $6.25.
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4. Suppose that by doubling the number of required units of nutritional element B from 60 to 120 for 2 weeks, the producer can realize $15 more from the sale of the stock than without the increase. Is this worthwhile?
an experiment consists of four outcomes with p(e1) = .2, p(e2) = .3, and p(e3) = .4. the probability of outcome e4 is _____.a. .900b. .100c. .024d. .500
The probability of outcome e4 is 0.1, which means the option (b). 0.100 is the correct answer.
To comprehend this response, keep in mind that the total probability for all outcomes in an experiment must equal 1. We now know the probability for e1, e2, and e3, which total 0.9 (0.2 + 0.3 + 0.4 = 0.9). Because the total of probabilities must equal one, we may remove 0.9 from one to get the chance of e4. As a result, the likelihood of e4 is 0.1 (1 - 0.9 = 0.1).
In other words, there are four possible outcomes in this experiment, with probabilities 0.2, 0.3, 0.4, and an unknown for e4. We may multiply the known probabilities by 0.9, leaving 0.1 for e4. This means that there is a 10% chance of outcome e4 occurring in this experiment.
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fill in the blank to complete the trigonometric identity. sin2(u) cos2(u) = tan2(u)
The trigonometric identity is sin²(u)/cos²(u) = tan²(u).
What is trigonometry?The study of the correlation between a right-angled triangle's sides and angles is the focus of one of the most significant branches of mathematics in history: trigonometry.
sin²(u) + cos²(u) = 1 is the trigonometric identity that relates the three basic trigonometric functions sine (sin), cosine (cos), and tangent (tan) of an angle u in a right-angled triangle.
However, to derive the identity sin²(u) / cos²(u) = tan²(u), we can start with the definition of tangent: tan(u) = sin(u) / cos(u).
Then, we can square both sides of the equation:
tan²(u) = (sin(u) / cos(u))²
tan²(u) = sin²(u) / cos²(u)
Therefore, sin²(u) / cos²(u) = tan²(u).
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The complete question is:
Fill in the blank to complete the trigonometric identity. sin²(u)__cos²(u) = tan²(u)
The point b is a reflection of point a across which axis?
Point b (7, 8) Point a (-7, -8).
A.The x-axis
B. The y-axis
C. The x-axis and then the y-axis
What is the area of a triangle, in square inches, with a base of 13 inches and a height of 10 inches
Answer: 65
Step-by-step explanation:
area of triangle = 1/2 of base * height
so 13*10 = 130
130 * 1/2 = 65
You need to buy a piece of canvas that is large enough to stretch and secure around a wooden frame. You plan that the length of your finished piece will be 5 inches less than twice the width, and you will need 2 inches extra on each side to secure the canvas to the frame. Which expression represents the area of the canvas?
A.) 2w^2+7w-4
B.) 2w^2+2w-5
C) 5w^2+7w-2
D.) 5w^2+3w-2
The area of the canvas can be calculated by taking the product of its length and width, which is 2w-5 and w+2, respectively, and then simplifying this expression to 5w²+3w-2, which is option D.
What is an expression?An expression is a mathematical statement that contains numbers, variables, and operations but lacks the equal sign.
To calculate the area of the canvas, we need to know the length and width of the canvas.
Since the length of the canvas will be 5 inches less than twice the width, the length can be expressed as:
2w-5.
The width of the canvas must include the extra 2 inches of fabric on each side to stretch and secure it around the frame, so the width is expressed as:
w+2.
The area of the canvas is then the product of the length and width,
= (2w-5)(w+2).
When this equation is simplified, it reduces to 5w²+3w-2, which is option D.
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if people are born with equal probability on each of the 365 days, what is the probability that three randomly chosen people have different birthdates?
The probability that three randomly chosen people have different birth date is 0.9918.
To calculate the probability that three randomly chosen people have different birthdates, we can first consider the probability that the second person chosen does not have the same birth day as the first person.
This probability is (364/365), since there are 364 possible birthdates that are different from the first person's birthdate, out of 365 possible birthdates overall.
Similarly, the probability that the third person chosen does not have the same birthdate as either of the first two people is (363/365), since there are now only 363 possible birthdates left that are different from the first two people's birthdates.
To find the overall probability that all three people have different birthdates, we can multiply these individual probabilities together:
(364/365) x (363/365) = 0.9918
So the probability that three randomly chosen people have different birth date is approximately 0.9918, or about 99.2%.
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The probability that three randomly chosen people have different birthdates is approximately 0.9918, or 99.18%.
The counting principle can be used to determine how many different birthdates can be selected from a pool of 365 potential dates. As we assume that people are born with equal probability on each of the 365 days of the year (ignoring leap years).
The first individual can be born on any of the 365 days. The second individual can be born on any of the remaining 364 days. The third individual can be born on any of the remaining 363 days. Therefore, the total number of ways to choose three different birthdates is:
365 x 364 x 363
Let's now determine how many different ways there are to select three birthdates that are not mutually exclusive (i.e., they can be the same). The number of ways to select three birthdates from the 365 potential dates is simply this:
365 x 365 x 365
Consequently, the likelihood that three randomly selected individuals have different birthdates is:
(365 x 364 x 363) / (365 x 365 x 365) ≈ 0.9918
Therefore, the likelihood is roughly 0.9918, or 99.18%.
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Suppose that f(x) = x/108 for 3 < x < 15. determine the mean and variance of x.
Round your answers to 3 decimal places. Mean = _____
Variance =____
Mean of the above function is 8.500 and the variance is 6.875.
To determine the mean and variance of x for the given function f(x) = x/108 for 3 < x < 15, we need to first calculate the mean and then the variance.
The mean, also known as the expected value, is the average value of a random variable. In this case, the random variable is x, and we need to find the expected value of x for the given function.
The integral of f(x) with respect to x from 3 to 15 gives us the expected value or the mean of x:
∫(x/108)dx from 3 to 15
= (1/108)∫xdx from 3 to 15 (using the power rule of integration)
= (1/108) * [(x^2)/2] from 3 to 15
= (1/108) * [(15^2)/2 - (3^2)/2]
= (1/108) * [(225/2) - (9/2)]
= (1/108) * (216/2)
= (1/108) * 108
= 1
So, the mean of x is 1.
Variance is a measure of how much the values of a random variable deviate from the mean. It is calculated as the average of the squared differences between the values and the mean.
The formula for variance is given by Var(x) = E[x^2] - E[x]^2, where E[x] is the expected value or the mean of x.
From the previous calculation, we know that E[x] = 1.
Now, we need to find E[x^2]. For this, we need to square the function f(x) and then find its expected value.
(f(x))^2 = (x/108)^2
= x^2 / 11664
The integral of (f(x))^2 with respect to x from 3 to 15 gives us the expected value of x^2:
∫(x^2/11664)dx from 3 to 15
= (1/11664)∫x^2dx from 3 to 15
= (1/11664) * [(x^3)/3] from 3 to 15
= (1/11664) * [(15^3)/3 - (3^3)/3]
= (1/11664) * [(3375/3) - (27/3)]
= (1/11664) * (3348/3)
= 0.286
Now, substituting the values of E[x^2] and E[x] into the formula for variance, we get:
Var(x) = E[x^2] - E[x]^2
= 0.286 - 1^2
= 0.286 - 1
= -0.714
So, the variance of x is -0.714.
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T/F - If A is an invertible n x n matrix, then the equation Ax = b is consistent for each b in R^n.
True, if A is an invertible n x n matrix, then the equation Ax = b is consistent for each b in Rⁿ.
When A is an invertible n x n matrix, it means that A has a unique inverse, denoted as A⁻¹, which is also an n x n matrix. This implies that for any given vector b in Rⁿ, there exists a unique solution x in Rⁿ that satisfies the equation Ax = b.
To understand why this is true, consider the definition of matrix multiplication. In the equation Ax = b, A is multiplied by x to obtain b. Since A is invertible, we can multiply both sides of the equation by A⁻¹ (the inverse of A) on the left, yielding A⁻¹Ax = A⁻¹b.
Now, according to the properties of matrix multiplication, A⁻¹A results in the identity matrix I_n (an n x n matrix with ones on the diagonal and zeros elsewhere), and any vector multiplied by the identity matrix remains unchanged. Therefore, we get I_nx = A⁻¹b, which simplifies to x = A⁻¹b.
This shows that for any given vector b in Rⁿ, there exists a unique solution x = A⁻¹b that satisfies the equation Ax = b when A is an invertible matrix. Hence, the equation Ax = b is consistent for each b in Rⁿ.
Therefore, the correct answer is True.
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let m be a positive integer. show that a mod m = bmodmifa≡b(modm).
To show that a mod m = b mod m if a ≡ b (mod m), we can use the definition of modular arithmetic.
To show that a mod m = b mod m if a ≡ b (mod m) for a positive integer m, we can follow these steps:
1. First, understand the given condition: a ≡ b (mod m) means that a and b have the same remainder when divided by the positive integer m. In other words, m divides the difference between a and b. Mathematically, this can be written as m | (a - b), which means there exists an integer k such that a - b = mk.
2. Next, recall the definition of modular arithmetic: a mod m is the remainder when a is divided by m, and similarly, b mod m is the remainder when b is divided by m.
3. We know that a - b = mk, so a = b + mk.
4. Now, divide both sides of the equation a = b + mk by m.
5. Since b and mk are both divisible by m, the remainder of this division will be the same for both sides. In other words, a mod m and b mod m have the same remainder when divided by m.
6. Therefore, we can conclude that a mod m = b mod m if a ≡ b (mod m) for a positive integer m.
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Use the given information to find the exact value of a. sin 2 theta, b. cos 2 theta, and c. tan 2 theta. cos theta = 21/29, theta lies in quadrant IV a. sin 2 theta =
The values we have found, we get:
a. sin(2theta) = 2(-20/29)(21/29) = -840/841
b. cos(2theta) = (21/29)² - (-20/29)² = 441/841 - 400/841 = 41/841
c. tan(2theta) = (2(-20/29))/(1 - (-20/29)²) = 40/9
What is trigonometry?
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
Since cos(theta) is positive and lies in quadrant IV, we know that sin(theta) is negative. We can use the Pythagorean identity to find sin(theta):
sin²(theta) + cos²(theta) = 1
sin²(theta) = 1 - cos²(theta)
sin(theta) = -sqrt(1 - cos²(theta))
Substituting cos(theta) = 21/29, we get:
sin(theta) = -sqrt(1 - (21/29)²) = -20/29
Now, we can use the double angle formulas to find sin(2theta), cos(2theta), and tan(2theta):
sin(2theta) = 2sin(theta)cos(theta)
cos(2theta) = cos²(theta) - sin²(theta)
tan(2theta) = (2tan(theta))/(1 - tan²(theta))
Substituting the values we have found, we get:
a. sin(2theta) = 2(-20/29)(21/29) = -840/841
b. cos(2theta) = (21/29)² - (-20/29)² = 441/841 - 400/841 = 41/841
c. tan(2theta) = (2(-20/29))/(1 - (-20/29)²) = 40/9
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A rock is dropped from a height of 100 feet. Calculate the time between when the rock was dropped and when it landed. If we choose "down" as positive and ignore air friction, the function is h(t) = 25t^2-81
at the movie theater, chil admission is $6.10 and adult admission is $9.40 on Friday, 136 tickets were sold for a total of $1027.60. how many adult tickets were sold that day?
Thus, the number of adult tickets that were sold that day was 60.
Explain about two variable linear equation:A linear equation with two variables is shown by the equation axe + by = r if a, b, and r are all real values and both are not equal to 0. The equation's two variables are represented by the letters x and y. The coefficients are denoted by the numerals a and b.
When the unknown variables in a polynomial equation have a degree of one, the equation is said to be linear. In other words, a linear equation's unknown variables are all increased to the power of 1.
For the given question:
Let the number of child tickets - x
Let the number of adult ticket - y
Price of each child tickets - $6.10
Price of each adult ticket- - $9.40
Thus, the system of linear equation forms -
x + y = 136
x = 136 - y ...eq 1
6.10x + 9.40y = 1027.60 ..eq 2
Put the value of x in eq 2
6.10x + 9.40y = 1027.60
6.10(136 - y) + 9.40y = 1027.60
829.6 - 6.10y + 9.40y = 1027.60
3.3y = 1027.60 - 829.6
3.3y = 198
y = 198/3.3
y = 60
x = 136 - 90
x = 46
Thus, the number of adult tickets that were sold that day was 60.
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which type of associations is a real relationship, not accounted by other variables?
A real relationship, not accounted by other variables, is a causal relationship. This type of association suggests that one variable directly causes changes in the other. In other words, there is a cause-and-effect relationship between the two variables.
In this type of association, changes in one variable directly cause changes in the other variable, without any other variables influencing the relationship. This contrasts with spurious or indirect associations, where the relationship between two variables is due to the influence of other variables. To determine if an association is a real relationship, researchers often control for potential confounding variables to isolate the direct effect of the variables in question. However, it is important to note that establishing a causal relationship requires careful research design and data analysis to rule out the effects of other variables that could be influencing the relationship.
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Electricity Company: Discount Power
Write a verbal description (word problem) for
this electricity company:
Complete the graph to represent the cost for
this electricity company. Choose appropriate
axis intervals and labels.
Complete the table to represent the cost for
this electricity company. Label each column
and choose the appropriate intervals.
Write an algebraic equation to represent the
costs for this electricity company
The word problem for the company is:
An electricity company charges its customers a monthly service fee of $3.50 plus 8.3 cents per kWH. Find the total cost for a month if 250kW is used.
What is a Word Problem?A word problem is a mathematical problem presented in the form of a story or a narrative, usually involving real-world scenarios or situations.
Word problems often require the use of arithmetic, algebra, geometry, or other mathematical concepts and methods to find a solution. They can range in complexity from simple arithmetic problems to multi-step equations involving multiple variables.
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Find the area of the cookie when the radius is 10 cm.
Use 3.14 for . If necessary, round your answer to the nearest hundredth.
The area of a cookie with radius of 10cm is given as follows:
A = 314 cm².
How to calculate the area of a circle?The area of a circle of radius r is given by the multiplication of π and the radius squared, as follows:
A = πr².
The cookie has a circular format, hence the equation used was presented above.
The radius is given as follows:
r = 10 cm.
Hence the area is given as follows:
A = 3.14 x 10²
A = 314 cm².
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The area of a cookie with radius of 10cm is given as follows:
A = 314 cm².
How to calculate the area of a circle?The area of a circle of radius r is given by the multiplication of π and the radius squared, as follows:
A = πr².
The cookie has a circular format, hence the equation used was presented above.
The radius is given as follows:
r = 10 cm.
Hence the area is given as follows:
A = 3.14 x 10²
A = 314 cm².
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