c = 1/2, and the pdf is fx(x) = { 1/2 [tex](1-x^{2} )^{-1}[/tex], −1 ≤ x ≤ 1or 0 otherwise and cdf of X is: FX(x) = { 0, x ≤ -1 or -1/2 [tex]tan^{-1}(x/\sqrt{1-x^{2} )}[/tex] + 1/2, -1 < x < 1 or 1 ,x ≥ 1
(a) To find c, we need to integrate the pdf over its support and set the result equal to 1 since the pdf must integrate to 1 over its entire support. Therefore, we have:
1 = ∫c [tex](1-x^{2} )^{-1}[/tex] dx from -1 to 1
Using the substitution u = 1 - [tex]x^{2}[/tex], we have:
1 = c∫ [tex]u^{-1/2}[/tex] dx from 0 to 1
Solving the integral, we get:
1 = 2c
Therefore, c = 1/2, and the pdf is:
fx(x) = {
1/2 [tex](1-x^{2} )^{-1}[/tex] , −1 ≤ x ≤ 1
0 otherwise
To sketch the pdf, we can notice that it is symmetric about x = 0 and that it approaches infinity as x approaches ±1. Therefore, it will have a peak at x = 0 and decrease as we move away from x = 0 in either direction.
(b) To find the cdf of X, we can integrate the pdf from negative infinity to x for each value of x in the support of the pdf. Therefore, we have:
FX(x) = ∫fX(t) dt from -∞ to x
For x ≤ -1, FX(x) = 0, since the pdf is zero for x ≤ -1. For -1 < x < 1, we have:
FX(x) = ∫1/2 [tex](1-t^{2} )^{-1}[/tex] dt from -1 to x
Using the substitution u = [tex]t^{2}[/tex] - 1, we have:
FX(x) = -1/2 ∫[tex](x^{2} -1)^{-1/2}[/tex] du
Solving the integral, we get:
FX(x) = -1/2 [tex]tan^{-1}(x/\sqrt{1-x^{2} )}[/tex] + 1/2
For x ≥ 1, we have FX(x) = 1, since the pdf is zero for x ≥ 1. Therefore, the cdf of X is:
FX(x) = {
0 x ≤ -1
-1/2 [tex]tan^{-1}(x/\sqrt{1-x^{2} )}[/tex] + 1/2 -1 < x < 1
1 x ≥ 1
To sketch the cdf, we can notice that it starts at 0 and increases gradually from x = -1 to x = 1, where it jumps to 1. The cdf is also symmetric about x = 0, similar to the pdf.
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When there is a problem with Solver being able to find a solution, many times it is an indication of a(n): ______
A. Older version of Excel
B. Mistakes in the formulation of the problem
C. Nonlinear programming problem
D. Problems that cannot be solved using linear programming
When there is a problem with Solver being able to find a solution, many times it is an indication of mistakes in the formulation of the problem. This means that the problem may not be correctly defined, or the constraints may not be properly specified.
However, it is also possible that the problem is a nonlinear programming problem, which can be more difficult for Solver to solve. In either case, it is important to carefully review the problem formulation and constraints to ensure that they are correct and accurately represent the problem at hand. It is also important to note that there may be some problems that cannot be solved using Solver or any other optimization tool, due to their inherent complexity or other factors.
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let p be the parallelogram determined by the vectors [4;1] and [3;-1]. let q be the shape obtained by applying the linear transformation t(x) = [3 1;1 2]x to the parallelogram p. fing the area of q. show all of your work.
The area of q is 20.
The area of a parallelogram determined by two vectors u and v is given by the magnitude of the cross product of u and v: |u x v|.
So, the area of the parallelogram p is:
| [4;1] x [3;-1] | = |(4)(-1) - (1)(3)| = |-7| = 7
To find the area of q, we apply the transformation T to each of the vertices of p and then compute the area of the resulting parallelogram.
First, we find the images of the vertices of p under T:
T([4;1]) = [3 1;1 2][4;1] = [16;6]
T([3;-1]) = [3 1;1 2][3;-1] = [6;1]
The sides of the parallelogram q are determined by the vectors T([4;1]) - T([3;-1]) = [10;5] and T([3;-1]) - [0;0] = [6;1].
The area of q is the magnitude of the cross product of these vectors:
| [10;5] x [6;1] | = |(10)(1) - (5)(6)| = |-20| = 20
Therefore, the area of q is 20.
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find the limit of the following sequence or determine that the sequence diverges. {tan^−1( 4n/ 4n +5)}
The limit of the given sequence is π/4, and the sequence converges to this value.
The given sequence is {tan^−1(4n/(4n+5))}. To determine if the sequence converges or diverges, we can analyze the limit of the function as n approaches infinity.
As n goes to infinity, the function behaves like tan^−1(4n/4n), which simplifies to tan^−1(1). Since the arctangent function has a range of (-π/2, π/2), tan^−1(1) falls within this range, and it is equal to π/4 (or 45° in degrees).
Now, let's consider the difference between the given function and the simplified one: (4n+5) - 4n = 5. As n becomes larger, the effect of the constant term 5 becomes negligible. Consequently, the function approaches tan^−1(1) as n approaches infinity.
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A company that teaches self-improvement seminars is holding one of its seminars in Somerville. The company pays a flat fee of $324 to rent a facility in which to hold each session. Additionally, for every attendee who registers, the company must spend $5 to purchase books and supplies. Each attendee will pay $32 for the seminar. Once a certain number of attendee register, the company will be breaking even. How many attendees will that take? What will be the company's total expenses and revenues?
For a company that wants to teach self improvement seminars is holding of its seminar. The company will be take 12 attendees to break even. The company's total expenses and revenues both are equal to $384.
We have a company that teaches self-improvement seminars is holding one of its seminars. Flat fee spent by company to rent a facility, P = $324
Additionally, Spent by company on books for every attendee who registers = $ 5
The fee pay by each attendee for attending the seminar = $32
We have to determine the number of attendees. Let 'x' represent the total number of attendees who are registered. According to the above scenario, the break-even equation is written as R (revenue) = E( expenses), 32x = 5 x + 324
Simplify it,
32x - 5x = 324
=> 27 x = 324
=> x = 324/27
=> x = 12
So, It will take 12 attendees to break even. Now, the company's total expenses
= 5x + 324
= 5×12 + 324
= $384
The net income or revenue will also be
= 32 ×12
= $384
Hence, required value is $384..
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solve the congruence 5x ≡ 11 and 11y ≡ 5, mod 37. if both are satisfied, simplify xy mod 37.
To solve a congruence like 5x ≡ 11 (mod 37), we need to find the modular inverse of 5 (mod 37) which is 15, then x ≡ 11(15) ≡ 27 (mod 37). Similarly, to solve 11y ≡ 5 (mod 37), we find the modular inverse of 11 (mod 37) which is 20, then y ≡ 5(20) ≡ 24 (mod 37).
The first step is to find the value of xy (mod 37). To do this, we simply multiply x and y and take the result modulo 37. For example, xy ≡ 27(24) ≡ 648 ≡ 11 (mod 37).
To solve a congruence like 5x ≡ 11 (mod 37), we need to find a number y such that 5y ≡ 1 (mod 37), since then we can multiply both sides of the congruence by y to get x ≡ 11y (mod 37). To find y, we use the extended Euclidean algorithm, which involves finding the greatest common divisor of 5 and 37 and expressing it as a linear combination of 5 and 37. In this case, we find that y = 15 is the modular inverse of 5 (mod 37), since 5(15) ≡ 1 (mod 37). Therefore, x ≡ 11(15) ≡ 27 (mod 37).
Similarly, to solve the congruence 11y ≡ 5 (mod 37), we need to find a number x such that 11x ≡ 1 (mod 37), since then we can multiply both sides of the congruence by x to get y ≡ 5x (mod 37). Using the extended Euclidean algorithm, we can find that x = 20 is the modular inverse of 11 (mod 37), since 11(20) ≡ 1 (mod 37). Therefore, y ≡ 5(20) ≡ 24 (mod 37).
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plot the point whose polar coordinates are given. then find the cartesian coordinates of the point. (C) -1, -π/6) . (X,Y)=
The point with polar coordinates (-1, -π/6) is plotted as a point on the terminal arm of an angle of -π/6 in the polar coordinate system. The Cartesian coordinates of the point are then determined using the relationships:
x = r cosθ and y = r sinθ, where r is the radius and θ is the angle in radians.
To find the Cartesian coordinates of the point, we substitute the given values for r and θ in the above equations:
x = (-1) cos(-π/6) = (-1) × (√3/2) = -√3/2
y = (-1) sin(-π/6) = (-1) × (-1/2) = 1/2
Therefore, the Cartesian coordinates of the point are (-√3/2, 1/2).
In summary, the point with polar coordinates (-1, -π/6) is plotted as a point on the terminal arm of an angle of -π/6 in the polar coordinate system. Then, using the relationships between polar and Cartesian coordinates, the Cartesian coordinates of the point are determined to be (-√3/2, 1/2).
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hannah invested $500 into an account with a 6.5% intrest rate compounded monthly. how much will hannahs investment be worth in 10 years.
Answer:
Using the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal (initial amount of investment)
r = the interest rate (as a decimal)
n = the number of times per year the interest is compounded
t = the time (in years)
Plugging in the values:
P = $500
r = 6.5% = 0.065
n = 12 (compounded monthly)
t = 10
A = 500(1 + 0.065/12)^(12*10)
A = $935.98
Hannah's investment will be worth $935.98 after 10 years.
Simplify the radical expression. Show all your steps.
√363 − 3√27
Answer: simplified expression is 2√3.
Step-by-step explanation:
√363 = √(121 × 3) = √121 × √3 = 11√3
√27 = √(9 × 3) = √9 × √3 = 3√3
√363 − 3√27 = 11√3 − 3(3√3) = 11√3 − 9√3 = 2√3
The simplified form of the given radical expression is 2√3.
What is radical form?Radical form is the expression that involves radical signs such as square root, cube root, etc instead of using exponents to describe the same entity.
The given expression is √363 − 3√27.
Here, √121×3 − 3√9×3
= 11√3-9√3
= 2√3
Therefore, the simplified form of the given radical expression is 2√3.
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what is the list after the second outer loop iteration?[6,9,8,1,7],[],,,
After the second outer loop iteration, the list is [6,1,7,8,9].
To determine the list after the second outer loop iteration, let's assume we're working with a simple bubble sort algorithm. Here are the steps:
1. First outer loop iteration:
- Compare 6 and 9; no swap.
- Compare 9 and 8; swap to get [6,8,9,1,7].
- Compare 9 and 1; swap to get [6,8,1,9,7].
- Compare 9 and 7; swap to get [6,8,1,7,9].
2. Second outer loop iteration:
- Compare 6 and 8; no swap.
- Compare 8 and 1; swap to get [6,1,8,7,9].
- Compare 8 and 7; swap to get [6,1,7,8,9].
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let f (x) = 5x and g(x) = x^1/3. find (fg) (x)
(fg)(x) =
The value of the function (fg)(x) = = ∛5
What is a function?A function can be described as an equation or expression that is used to show the relationship between two variables.
The two variables are;
The dependent variableThe independent variableFrom the information given, we have that;
f(x) = 5x
g(x) = x^1/3
To determine the composite function (fg)(x), substitute the value of(x) as the value of x in the function g(x), we have;
(fg)(x) = 5^1/3
This is written as;
(fg)(x) =(∛5)¹
(fg)(x) = = ∛5
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given f(7)=2, f′(7)=11, g(7)=−1, and g′(7)=9, find the values of the following. (a) (fg)′(7)= number (b) (fg)′(7)= number
Answer:
Step-by-step explanation:
Please help if you can, i don't understand
Answer: I believe -2 is the answer
Step-by-step explanation: To solve for the function over an interval, you need to know the equation of the function. If you have the equation, you can plug in the values of the interval into the equation to find the corresponding y-values. For example, if the function is y = 2x + 1 and the interval is [0,3], you can plug in x = 0 and x = 3 to find the corresponding y-values and get the ordered pairs (0,1) and (3,7).
What is the factored form of the polynomial?
x2 − 12x + 27?
(x + 4)(x + 3)
(x − 4)(x + 3)
(x + 9)(x + 3)
(x − 9)(x − 3)
Answer:
-9?
Step-by-step explanation:
Think About the Process A jar contains only pennies, nickels, dimes, and quarters. There are 18 pennies, 25 dimes, and 16 quarters. The rest of the coins are nickels. There are 88 coins in all. How many of the coins are not nickels? If n represents the number of nickels in the jar, what equation could you use to find n?
Around the beginning of the 1800’s, the population of the U.S. was growing at a rate of about 1.33^t million people per decade, with "t" being measured in decades from 1810.
If the population P(t) was 7.4 million people in 1810, estimate the population in 1820 (one decade later) by considering the work in example 2.
We can determine the population in 1820 was 8.5753 using a linear equation.
What does a linear equation mean in mathematics?A linear equation is one that has just a constant and a first order (linear) component, like y=mx+b, where m is the slope and b is the y-intercept.
When x and y are the variables, the aforementioned is sometimes referred to as a "linear equation of two variables."
dp/dt = [tex]1.37^{t}[/tex]
Integrate both sides.
p[h] = ( [tex]1.37^{t}[/tex])/In (1.37) + c
1810 ⇒ t = 0
7.4 = 1/In (1.37) + C
C = 4.2235
p(H) = ( [tex]1.37^{t}[/tex])/In (1.37) + 4.2235
P (1) = [tex]1.37^{t}[/tex]In (1.37) + 4.2235
= 8.5753
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A factory makes boxes of cereal. Each box contains cereal pieces shaped like hearts, stars,
and rings.
An employee at the factory wants to check the quality of a sample of cereal pieces from a box.
Which sample is most representative of the population?
Answer:
Step-by-step explanation:
d
The most representative sample of the population would be a random sample of cereal pieces from the box. Therefore, option D is the correct answer.
What is random sampling?In statistics, a simple random sample is a subset of individuals chosen from a larger set in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample in a random way.
Given that, a factory makes boxes of cereal. Each box contains cereal pieces shaped like hearts, stars and rings.
The most representative sample of the population would be a random sample of cereal pieces from the box. This means that the employee should select pieces from the box without looking at or attempting to select any particular shape. This ensures that the sample accurately reflects the distribution of cereal pieces in the box, and gives an accurate representation of the population.
Therefore, option D is the correct answer.
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in hypothesis testing, making decision that that causes a false alarm is equivalent to a. correct decision b. null hypothesis c. type-1 error d. type-2 error
In hypothesis testing, making a decision that causes a false alarm is equivalent to committing a type-1 error.
In hypothesis testing, making a decision that causes a false alarm is equivalent to a Type-1 error. This occurs when we reject the null hypothesis even though it is actually true. It is important to control the probability of making type-1 errors, as this can lead to incorrect conclusions and wasted resources. The correct decision in hypothesis testing is to either accept or reject the null hypothesis based on the evidence presented. A type-2 error, on the other hand, occurs when we fail to reject the null hypothesis even though it is false.
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Find the length of an arc of 40° in a circle with an 8 inch radius. 64 pi 1/9 inches
16 pi 1/9 inches
8 pi 1/9 inches
Answer:
16pi/9 in
Step-by-step explanation:
length of arc = (angle/360) x (2πr)
where angle is the central angle of the arc in degrees, r is the radius of the circle, and π is the mathematical constant pi (approximately equal to 3.14159).
In this case, the radius is given as 8 inches and the central angle is 40 degrees. Substituting these values into the formula, we get:
length of arc = (40/360) x (2π x 8)
length of arc = (1/9) x (16π)
length of arc = 16π/9
So the length of the arc is 16π/9 inches. Rounded to the nearest hundredth, this is approximately 5.60 inches. Therefore, the answer is 16 pi 1/9 inches, when expressed in mixed number form.
make a rectangle that’s x(x+1)=60 with the quadratic formula
The rectangle with the formula x(x+1)=60 using the quadratic formula has length = 8.26 and width = 7.26.
Given that,
A rectangle has the formula,
x (x + 1) = 60
x² + x = 60
x² + x - 60 = 0
Using the quadratic formula,
x = -1 ± √1 -(4 × 1 × -60) / 2
= (-1 ± √241) / 2
x = 7.26
Width = 7.26
Length = x + 1 = 8.26
Hence the required length and width are 8.26 and 7.26.
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we have g'(x) = 24x2 4x3. factoring this, we have: correct: your answer is correct. x2 correct: your answer is correct. x . thus, g'(x) = 0 when x = incorrect: your answer is incorrect. . (Enter your answers as a comma-separated list.)
The final answer is g'(x) = 0 when x = 0 or x = 6.
As an illustration, the phrase x + y is one where x and y are terms with an addition operator in between. There are two sorts of expressions in mathematics: numerical expressions, which only contain numbers, and algebraic expressions, which also include variables.
Given g'(x) = [tex]24x^2 - 4x^3[/tex], we need to find the value(s) of x such that g'(x) = 0. To do this, we factor the expression as follows:
[tex]g'(x) = 24x^2 - 4x^3 = 4x^2(6 - x)[/tex]
Setting g'(x) = 0, we have:
[tex]4x^2(6 - x) = 0[/tex]
This equation is satisfied when either [tex]4x^2 = 0[/tex]or 6 - x = 0. Solving for x, we get:
[tex]4x^2 = 0[/tex] => x = 0
6 - x = 0 => x = 6
Therefore, g'(x) = 0 when x = 0 or x = 6.
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Resuelve con proceso:
Un comerciante vende polos, 200 polos a 8 por 2 soles y 300 polos a 5 por 3 soles. ¿Cual es la diferencia de lo que recibió de la primera venta con la segunda?.
The number of more sole received by merchant in the second sale compared to first sale is equal to 130 soles
Let us first calculate the cost of one pole in each sale.
For the first sale, 8 poles cost 2 soles. So, one pole costs.
2 soles / 8 poles = 0.25 soles/pole
For the second sale, 5 poles cost 3 soles. So, one pole costs.
3 soles / 5 poles = 0.6 soles/pole
Next, let us find out the total revenue from each sale.
For the first sale,
The merchant sold 200 poles. If one pole costs 0.25 soles, then 200 poles would cost.
200 poles × 0.25 soles/pole = 50 soles
For the second sale,
The merchant sold 300 poles. If one pole costs 0.6 soles, then 300 poles would cost.
300 poles × 0.6 soles/pole = 180 soles
The difference between what the merchant received from the first sale and the second sale is,
180 soles - 50 soles = 130 soles
Therefore,, the merchant received 130 soles more from the second sale than the first sale.
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1. Solve the problem. If the price charged for a bolt is p cents, then x thousand bolts will be sold in a certain hardware store, where p=63-x/20. How many bolts must be sold to maximize revenue A) 630 thousand bolts B) 630 bolts C) 1260 bolts D) 1260 thousand bolts
A total of 630 thousand bolts must be sold to maximize revenue. The correct answer is A) 630 thousand bolts.
To maximize revenue, we need to first determine the revenue function.
Revenue is given by the product of price (p) and quantity (x).
In this case, p = 63 - x/20.
Write the revenue function:
R(x) = px
= (63 - x/20)x
Simplify the function:
R(x) = 63x - (x²)/20
To maximize the revenue, find the vertex of the parabola formed by the quadratic function.
The x-coordinate of the vertex is given by -b/(2a), where a and b are the coefficients of x² and x, respectively.
In this case, a = -1/20 and b = 63. So, the x-coordinate of the vertex is:
x = -63 / (2 (-1/20))
= 63 (20 / 2)
= 630.
Therefore, option A) is correct.
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find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y = a°, y = 25 about the r-axis.
The volume of the solid formed by rotating the region inside the first quadrant enclosed by y = a°, y = 25 about the r-axis is [tex]V = \pi(25^3 - (a^\circ)^3)[/tex].
To find the volume of the solid formed by rotating the region inside the first quadrant enclosed by y = a°, y = 25 about the r-axis, we can use the method of cylindrical shells.
First, we need to determine the limits of integration for y.
The region is enclosed by y = a° and y = 25, so the limits are a° and 25.
Next, we need to determine the radius of each cylindrical shell. Since we are rotating about the r-axis, the radius is simply the y-value.
So, the radius is r = y.
Finally, we need to determine the height of each cylindrical shell.
The height is the circumference of the shell, which is 2πr.
So, the height is h = 2πy.
The volume of each cylindrical shell is then given by V = 2πy * (y - a°)
To find the total volume, we integrate this expression with respect to y from a° to 25:
[tex]V = \int_{a^\circ}^{25} 2\pi (y - a^\circ) dy[/tex]
Evaluating this integral, we get:
[tex]V = \pi(25^3 - (a^\circ)^3)[/tex]
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Quickly answer please!
The graph of a function contains the points (-5, 1), (0,
3), (5, 5). Is the function linear? Explain.
(Photo of answer choice included)
(d) The function (-5, 1), (0, 3), (5, 5) is not a linear function
Calculating the type of the functionFrom the question, we have the following parameters that can be used in our computation:
(-5, 1), (0, 3), (5, 5).
A linear function has a constant rate of change, meaning that the slope of the line is always the same.
However, if we plot the given points on a graph, we can see that they do not lie on a straight line.
Therefore, the function is not linear.
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Evaluate the expression 7 + 2 x 8 − 5. (1 point)
18
20
48
63
therefore in the interval 0≤t≤6 t^2 6t-16 is negative when 0≤t≤
The expression t^2+6t-16 is negative in the interval 2≤t≤4 when 0≤t≤6.
To determine the interval where the expression t^2+6t-16 is negative, we need to solve the inequality t^2+6t-16<0. We can do this by factoring the quadratic expression or using the quadratic formula, but it's quicker to notice that the expression can be written as (t+4)(t-2)<0.
This means that the expression is negative when t is between -4 and 2, or when t is between 2 and 4, because the product of two factors is negative when one factor is negative and the other is positive. However, we are only interested in the interval between 0 and 6, so we need to check which of these subintervals satisfy that condition.
The subinterval between -4 and 2 is entirely outside the interval between 0 and 6, so we can ignore it. The subinterval between 2 and 4 is partially inside the interval between 0 and 6, but only the part between 2 and 4 is relevant. Therefore, the expression is negative when 2≤t≤4.
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I need HELP on all this questions
Answer:
1. The base is a square.
2. Its dimensions are 3 by 3 (or 3 x 3).
3. Their faces are triangles. (Not too sure if this correct as the wording of the question is a little confusing.)
4. The base of each triangle measures 3 cm.
5. The height of each triangle measures 4 cm.
6. The polyhedron is a cube.
Step-by-step explanation:
1. The question says to draw a net for the square pyramid. From there we already have our answer for the base.
2. It is a general rule that all sides of a square are equal. Hence, if one side is measured to be 3 cm, so will the rest.
3. This one is clear by sight, you can clearly recognize that the rest of the shapes are triangles. If not, you can also deduce this from the instructions saying that is a square pyramid. According to byjus.com, "A pyramid has a polygonal base and flat triangular faces, which join at a common point called the apex."
4. The side of the square is 3 cm. However, the triangle's base must be equal to this in order to form the pyramid and connect with the other triangles and square. We can also see this fact from the diagram, as the base of the triangle and the side of the square are the same.
5. The diagram points out that the height of the triangle is 4 cm. This is the only measurement left in the diagram, so it is likely to be the correct answer. Otherwise, this would be very difficult to solve.
6. If you contruct the net together, you will find that it forms a cube. We can also notice that there are 6 squares shown, and the cube is a 6-sided polyhedron. I have experience with forming paper cubes from my previous math classes, so I can confidently confirm that is a cube. If you still have doubts though, you may also research the net of a cube.
Hopefully this helped you out with your problem! I've answered this based from my own knowledge so please let me know if I misunderstood anything in the questions.
Help AGAIN!
Which one cheaper and by how much?
View attachment below
Answer: Website A is cheaper, by an amount of, £0.29.
Step-by-step explanation: Here, the problem is simply about, initially adding, and then finding difference between the added results.
That is,
For Website A,
Net Cost = £49.95 + £4.39
= £54.34
Similarly,
For Website B,
Net Cost = £47.68 + £6.95
= £54.63
Therefore, we can clearly see,
Website A is cheaper by,
£(54.63 - 54.34) = £0.29
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suppose x is a continuous variable with the following probability density: f(x)={c(10−x)2, if 0
Probability density function for the continuous variable x is:
f(x) = (3/1000)(10-x)², if 0
Total area under the probability density function is equal to 1.
So, we integrate the function from 0 to 10:
∫[0,10] c(10−x)2 dx
= c ∫[0,10] (10−x)2 dx
= c [-(10-x)³/³] evaluated from 0 to 10
= c [(0-(-1000/3))]
= c (1000/3)
Since the area under the probability density function is equal to 1, we have:
∫[0,10] c(10−x)2 dx = 1
Puting the value of the integral:
c (1000/3) = 1
Solving for c, we get:
c = 3/1000
Therefore, the probability density function for the continuous variable x is:
f(x) = (3/1000)(10-x)², if 0
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The characteristic polynomial of a 5 × 5 matrix is given below. Find the eigenvalues and their multiplicities.
λ^5 - 14λ^4 + 45λ^3
0 (multiplicity 1), 5 (multiplicity 1), 9 (multiplicity 1)
0 (multiplicity 3), 5 (multiplicity 1), 9 (multiplicity 1)
0 (multiplicity 3), -9 (multiplicity 1), -5 (multiplicity 1)
0 (multiplicity 1), -9 (multiplicity 1), -5 (multiplicity 1)
The eigenvalues of the 5 x 5 matrix are 0 (multiplicity: 3), 5 (multiplicity: 1), and 9 (multiplicity: 1).
Eigenvectors and eigenvalues have a large number of applications. The word eigenvalue means characteristic value. Therefore, the eigenvalues of the 5 x 5 matrix are represented by the roots of the characteristic polynomial. There is a term called multiplicity of an eigenvalue which is the number of times that a root is repeated.
Now, by algebraic means, we will determine the roots of the characteristic polynomial as follows:
[tex]p = λ^5 - 14λ^4 + 45λ^3\\p = λ^3( λ^2-14λ + 45)\\p = λ^3( λ^2 - 9λ - 5λ + 45)\\p = λ^3. (λ-5). (λ-9)[/tex]
The eigenvalues of the 5 x 5 matrix are 0 (multiplicity: 3), 5 (multiplicity: 1), 9 (multiplicity: 1)
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