The joint probability density function (pdf) of (X,Y ) is given by f(X,Y )(x,y) = 12/ 7 x(x + y), for 0 ≤ y ≤ 1, 0 ≤ x ≤ 1, 0, elsewhere. (a) Find the cumulative distribution function of (X,Y ). Make...
The joint probability density function (pdf) of (X,Y ) is given by f(X,Y )(x,y) = 12/ 7 x(x + y), for 0 ≤ y ≤ 1, 0 ≤ x ≤ 1, 0, elsewhere.
(a) Find the cumulative distribution function of (X,Y ). Make sure you derive expressions for the cdf in the regions • x < 0 or y < 0; • 0 ≤ x ≤ 1, 0 ≤ y ≤ 1; • x > 1, 0 ≤ y ≤ 1; • 0 ≤ x ≤ 1, y ≥ 1; • x > 1,y > 1.
(b) Evaluate (i) P(X > Y ); (ii) P(Y ≤ X2).
(c) Derive the pdf of X and then compute the mean and variance of X.
(d) Find the pdf of Y and compute the mean and variance of Y .
(e) Calculate the conditional pdf of Y given X = x.
(f) Compute the correlation coefficient ρ(X,Y ).
(g) Are X and Y independent? Explain.
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The joint probability density function (pdf) of (X,Y ) is given by f(X,Y )(x,y) = 12/ 7 x(x + y), for 0 ≤ y ≤ 1, 0 ≤ x ≤ 1, 0, elsewhere. (a) Find the cumulative distribution function of (X,Y ). Make...
How to get the cdf when y>x>0? Thanks 6. The joint probability density function (pdf) of...
How to get the cdf when y>x>0? Thanks
6. The joint probability density function (pdf) of (X, Y) is given by 0y<oo, elsewhere. fxr, y) (a) Find the cumulative distribution function of (X, Y) (b) Evaluate P(Y < X2) (c) Derive the pdf of X and then compute the mean and variance of X (d) Find the pdf of Y and compute the mean and variance of Y (e) Calculate the conditional pdf of Y given X (f) Compute the...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random var...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
Let (X,Y) have joint density f(x,y) -2x for0 <x < 1,0sys1 and 0 elsewhere. (a) Find...
Let (X,Y) have joint density f(x,y) -2x for0 <x < 1,0sys1 and 0 elsewhere. (a) Find P(xY > z) for 0szs1. Your final answer should be a function of z. (Hint: if you pick up a particular z, say,武what is the area within the unit square of 0 x 1 and 0 y 1 such that xy > z? P1.68 shows what you need to do, i.e., a double integral. Note thatz is a constant from the perspective of both...
The joint probability density function (PDF) of random variables X and Y is given by: f(x,y)...
The joint probability density function (PDF) of random variables X and Y is given by: f(x,y) = 4xy for 0 ≤ y ≤ x ≤ 1, and = 0 elsewhere The mean of the random variable X is:
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y)...
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
Assume that the joint density function of X and Y is given by f (x, y)...
Assume that the joint density function of X and Y is given by f (x, y) = 4,0 < x < 2,0 < y = 2 and f (x, y) = 0 elsewhere. (a) Find P (X < 1, Y > 1). (b) Find the joint cumulative distribution function F(x, y) of the two random variables. Include all the regions. (c) Find P (X<Y). (d) Explain how the value of P (1 < X < 2,1 < Y < 2)...
7. Given the joint density function /(x,y) =(kx (1 + 3 y*) 0<x<2,0<p?1 elsewhere a. Find...
7. Given the joint density function /(x,y) =(kx (1 + 3 y*) 0<x<2,0<p?1 elsewhere a. Find k, g() h) and f(x) b. Evaluate P(-<X<1)
0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then fi...
0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then find the constant k and sketch f(r.y) in three dimensions (b) Find and sketch the marginal pdf fx), the conditional pdf(x1/2) and the conditional cdf FO11/2). Find P(X〈Y! Y〉 1/2), E(XİY=1/2) and E(XIY〉l/2). (c) What is the correlation between X...
The following joint probability distribution is given. 1. Find k such that the given function demonstrates...
The following joint probability distribution is given. 1. Find k
such that the given function demonstrates the PDF. 2. Find Marginal
distributions. 3. Evaluate ?(? < ? < 0) 4. Find the
correlation coefficient between X and Y having the joint density
functions:(.) ?(?,?) = {???2+?2 ??? ?2 + ?2 < 4 0
?????h???
Question 2. (20 pts.) The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions....
Let X1 and X2 have joint PDF f(x1,x2)=x1+x2 for 0 <x1 <1 and 0<x2 <1.(a) Find...
Let X1 and X2 have joint PDF f(x1,x2)=x1+x2 for 0 <x1 <1 and 0<x2 <1.(a) Find the covariance and correlation of X1 and X2. (b) Find the conditional mean and conditional variance of X1 given X2 = x2.
How to get the cdf when y>x>0? Thanks
6. The joint probability density function (pdf) of (X, Y) is given by 0y<oo, elsewhere. fxr, y) (a) Find the cumulative distribution function of (X, Y) (b) Evaluate P(Y < X2) (c) Derive the pdf of X and then compute the mean and variance of X (d) Find the pdf of Y and compute the mean and variance of Y (e) Calculate the conditional pdf of Y given X (f) Compute the...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
Let (X,Y) have joint density f(x,y) -2x for0 <x < 1,0sys1 and 0 elsewhere. (a) Find P(xY > z) for 0szs1. Your final answer should be a function of z. (Hint: if you pick up a particular z, say,武what is the area within the unit square of 0 x 1 and 0 y 1 such that xy > z? P1.68 shows what you need to do, i.e., a double integral. Note thatz is a constant from the perspective of both...
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
Assume that the joint density function of X and Y is given by f (x, y) = 4,0 < x < 2,0 < y = 2 and f (x, y) = 0 elsewhere. (a) Find P (X < 1, Y > 1). (b) Find the joint cumulative distribution function F(x, y) of the two random variables. Include all the regions. (c) Find P (X<Y). (d) Explain how the value of P (1 < X < 2,1 < Y < 2)...
7. Given the joint density function /(x,y) =(kx (1 + 3 y*) 0<x<2,0<p?1 elsewhere a. Find k, g() h) and f(x) b. Evaluate P(-<X<1)
0 〈 y 〈 x2く1· Consider two rvs X and Y with joint pdf f(x,y) = k-y, (a) Sketch the region in two dimensions where fx,y) is positive. Then find the constant k and sketch ) in three imesions Then find the constant k and sketch f(r.y) in three dimensions (b) Find and sketch the marginal pdf fx), the conditional pdf(x1/2) and the conditional cdf FO11/2). Find P(X〈Y! Y〉 1/2), E(XİY=1/2) and E(XIY〉l/2). (c) What is the correlation between X...
The following joint probability distribution is given. 1. Find k
such that the given function demonstrates the PDF. 2. Find Marginal
distributions. 3. Evaluate ?(? < ? < 0) 4. Find the
correlation coefficient between X and Y having the joint density
functions:(.) ?(?,?) = {???2+?2 ??? ?2 + ?2 < 4 0
?????h???
Question 2. (20 pts.) The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions....
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