Consider the following continuous Joint PDF. х f(x,y) = K. x/y^2 In 1/y + < x < y and 1 a. Sketch the region where the PDF lies. b. Find the value of the constant K that makes this a valid joint probability density function c. Find the marginal density function of Y.
Answers
a. The given joint PDF is defined as follows:
f(x, y) = K * (x / [tex]y^2[/tex]) * (1/y), for 1 < x < y and 1 < y.
b. The value of the constant K that makes this a valid joint PDF is K = 2y.
c. The marginal density function of Y is [tex]f_{Y(y)}[/tex] = 1 - (1/[tex]y^2[/tex]).
To analyze the continuous joint probability density function (PDF) provided, we can follow these steps:
a. Sketching the region where the PDF lies:
The given joint PDF is defined as follows:
f(x, y) = K * (x / [tex]y^2[/tex]) * (1/y), for 1 < x < y and 1 < y.
To sketch the region, we can visualize the bounds of x and y based on the conditions given. The region lies within the range where x is between 1 and y, and y is greater than 1. This can be represented as follows:
Note: Find the attached image for the sketched region.
The region lies above the line y = 1, with x bounded by the lines x = 1 and x = y.
b. Finding the value of the constant K:
For the given function to be a valid joint probability density function, the integral of the joint PDF over the entire region must equal 1. Mathematically, we need to find the constant K that satisfies the following condition:
∫∫ f(x, y) dx dy = 1
The integral is taken over the region where the PDF lies, as determined in part (a). To find the constant K, we integrate the PDF over the given region and set it equal to 1. The integral can be taken as follows:
∫∫ f(x, y) dx dy = ∫∫ K * (x / [tex]y^2[/tex]) * (1/y) dx dy
Integrating with respect to x first, and then y, we have:
∫(y to ∞) ∫(1 to y) K * (x / [tex]y^2[/tex]) * (1/y) dx dy = 1
Simplifying the integral:
K * (1/y) ∫(y to ∞) [x] (1/[tex]y^2[/tex]) dx dy = 1
K * (1/y) [([tex]x^2[/tex] / (2 * [tex]y^2[/tex]))] (y to ∞) = 1
K * (1/y) * [([tex]y^2[/tex] / (2 * [tex]y^2[/tex]))] = 1
K * (1/y) * (1/2) = 1
Solving for K:
K = 2y
Therefore, the value of the constant K that makes this a valid joint PDF is K = 2y.
c. Finding the marginal density function of Y:
To find the marginal density function of Y, we integrate the joint PDF f(x, y) over the entire range of x, while considering y as the variable of interest.
Mathematically, the marginal density function of Y, denoted as [tex]f_{Y(y)}[/tex], can be computed as follows:
[tex]f_{Y(y)}[/tex] = ∫ f(x, y) dx
Integrating the joint PDF f(x, y) with respect to x, we have:
[tex]f_{Y(y)}[/tex] = ∫(1 to y) K * (x / [tex]y^2[/tex]) * (1/y) dx
Simplifying the integral:
[tex]f_{Y(y)}[/tex] = K * (1/y) ∫(1 to y) (x / [tex]y^2[/tex]) dx
[tex]f_{Y(y)}[/tex] = K * (1/y) [([tex]y^2[/tex] / (2 * [tex]y^2[/tex]))] (1 to y)
[tex]f_{Y(y)}[/tex]= K * (1/y) * [(([tex]y^2[/tex] / (2 * [tex]y^2[/tex])) - (1^2 / (2 * [tex]y^2[/tex])))]
[tex]f_{Y(y)}[/tex]= K * (1/y) * [(1/2) - (1/2[tex]y^2[/tex])]
Substituting the value of K = 2y, we get:
[tex]f_{Y(y)}[/tex]= 2y * (1/y) * [(1/2) - (1/2[tex]y^2[/tex])]
Simplifying further:
[tex]f_{Y(y)}[/tex]= 1 - (1/[tex]y^2[/tex])
Therefore, the marginal density function of Y is [tex]f_{Y(y)}[/tex] = 1 - (1/[tex]y^2[/tex]).
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Related Questions
why why why do i have to watch ads
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Answer:
Because you didnt pay for brainly.
If you do, no adds :D
Its a lot easier to be honest
Answer: Because the advertisements are corporations' way of making a market with the internet. They pay a fee to have their adds placed on sites and above videos, and by watching them, that generates revenue for the content creator and the hosting domain itself.
you welcome
Step-by-step explanation:
5. use the integral representation of Jy(x) 1 XV Jv(x) = tixp (1 – p2)v-żdp (+1,0T ©) p VT(v-3): 1 - To show that the spherical Bessel functions in (x) are expressible in terms of trigonometric functions, that is, for example, sin x sin x j.(x) = x ji(x) х x2 COS X = = ) х
Answers
We are given a formula that uses the integral representation of Jy(x): $${x\over 2}\left[J_{v-1}(x) - J_{v+1}(x)\right] = v\int_0^\infty t^{v-1} J_{v-1}(xt)\,dt$$
We will use this formula to show that the spherical Bessel functions $j _v(x)$ are expressible in terms of trigonometric functions. Let $y=v-1$.
Substituting $v=y+1$, we have: $${x\over 2}\left[J_y(x) - J_{y+2}(x)\right] = (y+1)\int_0^\infty t^y J_y(xt)\,dt$$
This expression is known as a recurrence relation for spherical Bessel functions. Let us use this to derive the identity that was requested:
$${x\over 2}\left[J_{v-1}(x) - J_{v+1}(x)\right] = (v+1)\int_0^\infty t^v J_v(xt)\,dt - v\int_0^\infty t^v J_{v-1}(xt)\,dt$$
Rearranging and using the recurrence relation, we obtain:
$$J_{v+1}(x) = {2v\over x}J_v(x) - J_{v-1}(x)$$$$\begin{aligned}\sin x\,j_v(x) &= \sin x\left[{1\over x}J_v(x)\right]\\&= {1\over 2x}\left[J_{v-1}(x) - J_{v+1}(x)\right]\\&= {v+1\over x}\int_0^\infty t^v J_v(xt)\,dt - {v\over x}\int_0^\infty t^v J_{v-1}(xt)\,dt\end{aligned}$$
Similarly, $$x^2\cos x\,j_v(x) = {v\over x}\int_0^\infty t^v J_v(xt)\,dt + {v+1\over x}\int_0^\infty t^v J_{v-1}(xt)\,dt$$
Hence, we have shown that $j_v(x)$ can be expressed in terms of trigonometric functions as follows:$$\begin{aligned}\sin x\,j_v(x) &= {v+1\over x}\int_0^\infty t^v J_v(xt)\,dt - {v\over x}\int_0^\infty t^v J_{v-1}(xt)\,dt\\\cos x\,j_v(x) &= {v\over x}\int_0^\infty t^v J_v(xt)\,dt + {v+1\over x}\int_0^\infty t^v J_{v-1}(xt)\,dt\end{aligned}$$
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write five other iterated integrals that are equal to the iterated integral
∫¹₀ ∫¹ᵧ ∫ʸ ₀ f(x, y, z)dx dx dy
Answers
Here are five other iterated integrals that are equal to the given iterated integral:
∫₀ʸ ∫⁺∞ₓ ∫¹₀ f(x, y, z)dz dx dy
∫₀ʸ ∫¹₀ ∫ʸ ∞ f(x, y, z)dz dx dy
∫⁺∞₁₀ ∫₀ʸ ∫ʸ ₀ f(x, y, z)dz dy dx
∫¹ᵧ ∫⁺∞₁₀ ∫₀ x f(x, y, z)dz dx dy
∫⁺∞₀ ∫⁺∞₁₀ ∫ʸ ₀ f(x, y, z)dz dx dy
The given iterated integral ∫¹₀ ∫¹ᵧ ∫ʸ ₀ f(x, y, z)dx dx dy represents the integration of a function f(x, y, z) over a region defined by the limits of integration. To obtain five other equivalent iterated integrals, we can rearrange the order of integration and modify the limits accordingly. Each integral represents the same volume or value as the given iterated integral, but the order of integration and limits may vary.
The key is to ensure that the new integrals cover the same region as the original one. The limits in each integral should define the appropriate range for each variable to maintain the equivalence. By rearranging the order of integration and adjusting the limits accordingly, we can obtain these alternative expressions that are equal to the given iterated integral.
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Write the equation of the given line in slope-intercept form:
(-1,2) and (1,-4)
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answer is y=-3x+(-1/3)
Is 9.90 equal to greater than or less than 9.9
Answers
Answer:
Equal to.
Step-by-step explanation:
The 0 at the end of 9.90 does not add any value to the number because its 0.
Answer:
It is equal.
Step-by-step explanation:
9.90 = 9.9 , it's the same as the 0 does not count on it.
how many strings of length 5 are there over the alphabet {0, 1, 2}?
Answers
The length of the string is 5 and the alphabet is {0, 1, 2}.Therefore, the number of strings of length 5 that can be formed over the given alphabet is:$$3^5 = 243$$ Therefore, there are 243 strings of length 5 over the alphabet {0, 1, 2}.
To calculate the number of strings of length 5 over the alphabet {0, 1, 2}, we need to determine the number of choices for each position in the string. Since each position can be filled with one of three possible characters (0, 1, or 2), we have three choices for each position.
Therefore, the total number of strings of length 5 can be calculated as:
Number of strings = Number of choices for position 1 × Number of choices for position 2 × Number of choices for position 3 × Number of choices for position 4 × Number of choices for position 5
Number of strings = 3 × 3 × 3 × 3 × 3 = 3^5 = 243
So, there are 243 strings of length 5 over the alphabet {0, 1, 2}.
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The information given in question is that the length of the string is 5.
Alphabet {0,1,2}
Therefore, there are 243 strings of length 5 over the alphabet {0, 1, 2}.
To find the number of strings of length 5 over the alphabet {0, 1, 2}, we need to consider the number of choices we have for each position in the string.
There are three choices (0, 1, or 2) for each position, and since we have five positions, the total number of strings of length 5 is given by:
[tex]$$3^5 = \boxed{243}$$[/tex]
Therefore, there are 243 strings of length 5 over the alphabet {0, 1, 2}.
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Slove the system of linear equations by graphing y=-x+7 y=x-1
Answers
Answer: Point Form: ( 4 , 3 ) Equation Form: x = 4 , y = 3
Step-by-step explanation: Solve for the first variable in one of the equations, then substitute the result into the other equation.
Brainliest or a thank you please? :))
I’ll give brainless too who ever respond fast and correctly.
Answers
Rachel has 42 stickers she wants give to her friends. Rachel put the stickers in 7 equal groups. Then Rachel found more stickers and put 3 more in each pile. If Rachel gives 5 stickers to each of her 12 friends, how many stickers does she have left over?
Answers
Answer:
3 stamps
Step-by-step explanation: 42 stamps in equal groups would be 6 stickers/group.
6+3 = 9 stickers in each pile
there was 7 piles so multiply 7 by the 9 stickers in each pile
7· 9 = 63
5 · 12 = 60, so she gave her friends 60 stickers
63-60 = 3
Of all the animals admitted to the local pet hospital, 10 of them are dogs. They represent 25% of all the animals at the hospital. How many animals are at the hospital in total? A. 20 B. 30 C. 40 D. 50
Answers
Answer:
C. 40
Step-by-step explanation:
[tex]\frac{25}{100} = \frac{10}{y}[/tex]
[tex]\frac{1}{4} = \frac{10}{y}[/tex] (cross multiply)
40 = 1y (rewrite)
y = 40
Find all the first derivatives of the function f(x,y) = x^0.9y^1.8. Show all of your steps with explanations of what you are doing.
Answers
The first derivative of the given function with respect to x is 0.9x^(-0.1)y^1.8 and the first derivative of the given function with respect to y is 1.8x^0.9y^0.8.
The given function is f(x, y) = x^0.9y^1.8. The question is to find the first derivative of the given function with respect to x and y respectively.
Here are the solutions; The first derivative of the given function with respect to x is given by ∂f(x, y)/∂x:∂f(x, y)/∂x = 0.9x^(-0.1)y^1.8
On the other hand, the first derivative of the given function with respect to y is given by ∂f(x, y)/∂y:∂f(x, y)/∂y = 1.8x^0.9y^(1.8 - 1)∂f(x, y)/∂y = 1.8x^0.9y^0.8
Therefore, the first derivative of the given function with respect to x is 0.9x^(-0.1)y^1.8 and the first derivative of the given function with respect to y is 1.8x^0.9y^0.8.
The above steps use the concept of partial derivatives of a function with respect to a variable that can be applied in such types of questions.
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The given function is
[tex]f(x,y) = x⁰.⁹ y¹.⁸[/tex]
Now, let's find the first derivative of the function with respect to x and y.
First derivative of the function with respect to x:
We have to use the chain rule here.
According to the chain rule,
the derivative of the outer function is multiplied by the derivative of the inner function.
The inner function is [tex]y¹.⁸,[/tex]
whose derivative with respect to x is 0.
Therefore, we only have to differentiate the outer function with respect to x.
[tex]f(x,y) = x⁰.⁹ y¹.⁸∂f/∂x = ∂/∂x (x⁰.⁹ y¹.⁸)∂f/∂x = 0.⁹x^(-0.1) y¹.⁸∂f/∂x = 0.⁹y¹.⁸/x^0.1[/tex]
First derivative of the function with respect to y:
We have to use the chain rule here.
According to the chain rule, the derivative of the outer function is multiplied by the derivative of the inner function.
The inner function is [tex]x⁰.⁹,[/tex]
whose derivative with respect to y is 0.
Therefore, we only have to differentiate the outer function with respect to y.
[tex]f(x,y) = x⁰.⁹ y¹.⁸∂f/∂y = ∂/∂y (x⁰.⁹ y¹.⁸)∂f/∂y = 1.⁸ x⁰.⁹ y^(0.8)∂f/∂y = 1.⁸x^0.9 y^0.8[/tex]
Hence, the first derivative of the given function with respect to x is
[tex]0.⁹y¹.⁸/x^0.1[/tex]
and with respect to y is [tex]1.⁸x^0.9 y^0.8.[/tex]
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Identify if they are function or not
Answers
Answer:
1=function
2=not function
3=function
4=function
5=function
6=not function
7=not function
8=function
Determine if the following linear maps are surjective or injective. You may assume each map is linear. (a) The derivative map T : P3(R) → P₂(R); P₂ (R); p(x) → d dx -p(x). (b) The linear map T: R³ → Rª defined by the matrix 1 1 -3 1 1 1 A 0 1 2 1 0 -1 (c) The linear map T: R³ → R³ defined by the matrix 0 5 5 A = 2 1 2 2 3 4
Answers
The linear map is not surjective but injective.
A linear map is an operation between two vector spaces that preserves the properties of addition and scalar multiplication; therefore, it is a linear transformation.
Here is how to determine if the following linear maps are surjective or injective.
(a) The derivative map T: P3(R) → P₂(R); P₂(R); p(x) → d/dx -p(x) is an example of a linear map,
where P3(R) and P₂(R) are the vector spaces of polynomials of degree at most 3 and 2 with coefficients in R, respectively.
Moreover, d/dx is the derivative operator acting on the polynomial p(x).
The kernel of the linear map T is the subspace of P3(R) consisting of polynomials p(x) with T(p(x)) = d/dx -p(x) = 0, i.e., p(x) is constant.
However, a constant polynomial of degree zero is not in the range of T, since it has no derivative. Therefore, the linear map T is injective and not surjective.
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Use the distributive property to rewrite each algebraic expression 7(y+2) + (8+r) + 8(x + 9)
Answers
Answer:
7y+r+8x+94
Explanation
Given the algebraic expression 7(y+2) + (8+r) + 8(x + 9)
Given A, B and C. According to distributive law;
A(B+C) = AB + AC
A is distributed over C. In the same vein
On expanding the expression;
7(y+2) + (8+r) + 8(x + 9)
7y+7(2) + 8 + r + 8(x)+ 8(9)
7y+14+8+r+8x+72
Bringing the variables and constants together
7y+r+8x+14+8+72
7y+r+8x+94
PLS ANSWER THIS AND GET BRAINLIEST !!!
A grocery store collected sales data. It found that when customers buy less bread, they tend to purchase more rice. What can we conclude?
A. There is no correlation between amount of bread bought and amount of rice purchased.
B. There is a correlation between amount of bread bought and amount of rice purchased. However, there is no causation. This is because there is an increase in the amount of rice purchased with a decrease in the amount of bread bought.
C. There is a correlation between amount of bread bought and amount of rice purchased. There may or may not be causation. Further studies would have to be done to determine this.
Answers
Answer: C
Step-by-step explanation: C because there is obviously a correlation yet we cannot determine if there is causation or not.
How long will it take for quarterly deposits of $625 to accumulate to be $20,440 at an interest rate of 8.48% compounded quarterly?
Answers
It will take approximately 9 years and 2 months for quarterly deposits of $625, with an interest rate of 8.48% compounded quarterly, to accumulate to $20,440.
To calculate the time it takes for the deposits to accumulate to the desired amount, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the accumulated amount
P = the principal amount (initial deposit)
r = the annual interest rate (converted to a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the principal amount (P) is $625, the interest rate (r) is 8.48% (or 0.0848 as a decimal), the number of times interest is compounded per year (n) is 4 (quarterly compounded), and the desired accumulated amount (A) is $20,440.
We need to solve for t, the number of years. Rearranging the formula, we have:
t = (log(A/P)) / (n * log(1 + r/n))
Plugging in the values, we get:
t = (log(20440/625)) / (4 * log(1 + 0.0848/4))
Calculating this, we find that t is approximately 9.18 years. Converting this to years and months, we get approximately 9 years and 2 months. Therefore, it will take around 9 years and 2 months for the quarterly deposits of $625 to accumulate to $20,440 at an interest rate of 8.48% compounded quarterly.
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What was different about the Yuan Dynasty? *
I’ll give brainiest answer to the person who gets it right.
Answers
Answer:
One big change during Kublai's reign was that foreigners became the rulers and administrators. Since they didn't trust the local people, they moved in a large number of Muslims and other people to help them rule the empire. The Mongols had their own religious belief called Shamanism.
Solve for w. Make sure to use scrap paper to show your work. ( 36 ÷ 3 )( 2 x 6 ) = w
Answers
Answer:
your answer is w=144
Step-by-step explanation:
(36÷3=12)(2x6=12)
12x12=144Answer: 144
Step-by-step explanation:
hey lol pls help <3
Ms. Philor is going on a trip to Hawaii. The function A(d)=0.80d+200 models the amount A, in dollars, that Ms.Philor's company pays her based on the round trip distance d, in miles that Ms. Philor travels to do a job.
Ms. Philor's pay increases by $
A)80.200
B) 0.80
C)200.80
Answers
Sarah practices the piano 1/3 hour in the morning, 5/6 hour in the afternoon, and 4/5 hour in the evening.
How many hours did Sarah practice in all?
Answers
4. Write neatly and legibly.
QUESTION 1
1.1
Find the sum:
1.1.1 2+6+(-7) + 10 = 11
Answers
Find the sum:
1.1.1 2+6+(-7) + 10 = 11
11=11
So first you need to remove the brackets
When there is a in front of an expression in brackets, the expression remains the same
2+6-7+10=11
Then combine the numbers and it comes out as 11 so it’s true
PLEASE HURRY!! I need help!!!
Calculate the IQR given in the box: 
Answers
solve elimination method 3x+4y=0 and x - 2y =-5
Answers
Answer:
x=−2 and y=3/2
Step-by-step explanation:
see picture
x h(x)
-25 6
-13 0
-3 -5
0 -7
9 -11
11 -30
What is the average rate of change of h(x) over the interval -13 < x < 11 ?
Answers
Answer:
56
Step-by-step explanation:
if 1-3(y+2)=-32 find the value of y/3
Answers
Answer:
7.6
Step-by-step explanation:
-3 multiple by y and -3 multiple by 2 resolve to 7.6....
Based on the two data sets represented below, complete the following sentences.
\text{DATA SET B}
DATA SET B
0
0
23
24
25
26
27
28
29
30
31
32
33
\text{DATA SET C}
DATA SET C
0
0
23
24
25
26
27
28
29
30
31
32
33
The median of Data Set B is
than the median of Data Set C. The minimum of Data Set B is
than the minimum of Data Set C.
Answers
Investigators performed a randomized experiment in which 411 juvenile delinquents were randomly assigned to either multisystemic therapy (MST) or just probation (control group). Of the 215 assigned to therapy, 87 had criminal convictions within 12 months. Of the 196 in the control group, 74 had criminal convictions within 12 months. Determine whether the therapy caused significantly fewer arrests at a 0.05 significance level. Start by comparing the sample percentages. Find and compare the sample percentages that were arrested for these two groups. The percentage of arrests for people who received MST was %.
Answers
The percentage of arrests for people who received Multisystemic Therapy (MST) can be calculated by dividing the number of individuals arrested in the MST group (87) by the total number of individuals.
Percentage of arrests for MST group = (87/215) * 100 ≈ 40.47%
To determine if therapy caused significantly fewer arrests at a 0.05 significance level, we need to compare this percentage with the percentage of arrests in the control group.
The percentage of arrests for the control group can be calculated in a similar manner by dividing the number of individuals arrested in the control group (74) by the total number of individuals in the control group (196), and multiplying by 100.
Percentage of arrests for control group = (74/196) * 100 ≈ 37.76%
Comparing the sample percentages, we find that the percentage of arrests for people who received MST (40.47%) is slightly higher than the percentage of arrests for the control group (37.76%).
To determine if this difference is statistically significant at a 0.05 significance level, we would need to perform a hypothesis test, such as a chi-square test, to compare the observed frequencies with the expected frequencies under the assumption that therapy has no effect on reducing arrests.
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how do I graph this linear equation?
4x+6y=-12
Answers
Answer: Simplifying
Step-by-step explanation:
Simplifying
4x + 6y = 12
Solving
4x + 6y = 12
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-6y' to each side of the equation.
4x + 6y + -6y = 12 + -6y
Combine like terms: 6y + -6y = 0
4x + 0 = 12 + -6y
4x = 12 + -6y
Divide each side by '4'.
x = 3 + -1.5y
Simplifying
x = 3 + -1.5y
Then you get your answer I think or hopeful get your answer. I hoped this helped!
data analysts use one-sample hypothesis tests to ______. a. frustrate students b. manage random sampling error c. justify their jobs d. make bad decisions e. all of the choices above f. none of the choices
Answers
Data analysts use one-sample hypothesis tests to manage random sampling error. The correct answer is (b).
Data analysts use one-sample hypothesis tests to manage random sampling error. Random sampling error refers to the variability or differences that can occur between a sample and the population it represents.
By conducting hypothesis tests, analysts can determine if the observed data from a sample is statistically significant and can be generalized to the larger population.
Hypothesis tests help analysts assess whether an observed effect or relationship in the sample is likely to be a true effect or relationship in the population or if it is simply due to random chance.
By testing a hypothesis and comparing the sample data to a null hypothesis, analysts can evaluate the validity of their findings and make informed decisions based on the results.
The purpose of hypothesis tests is not to frustrate students, justify their jobs, or make bad decisions. Instead, they serve as a statistical tool to manage random sampling error and provide reliable and valid conclusions based on data analysis.
The correct answer is (b)
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plss help answer true or false
Answers
Answer:
"false" I believe if wrong srry