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• ### 4. Two random variables X and Y have the following joint probability density function (PDF) Skx 0<x<y<1, fxy(x...

4. Two random variables X and Y have the following joint probability density function (PDF) Skx 0<x<y<1, fxy(x, y) = 10 otherwise. (a) [2 points) Determine the constant k. (b) (4 points) Find the marginal PDFs fx(2) and fy(y). Are X and Y independent? (c) [4 points) Find the expected values E[X] and EY). (d) [6 points) Find the variances Var[X] and Var[Y]. (e) [4 points) What is the covariance between X and Y?

• ### A) Let X and Y be two random variables with known joint PDF Ir(x, y). Define two new random varia...

a) Let X and Y be two random variables with known joint PDF Ir(x, y). Define two new random variables through the transformations W=- Determine the joint pdf fz(, w) of the random variables Z and W in terms of the joint pdf ar (r,y) b) Assume that the random variables X and Y are jointly Gaussian, both are zero mean, both have the same variance ơ2 , and additionally are statistically independent. Use this information to obtain the joint...

• ### 7. Two random variables X and Y have joint probability density function s(x, y) = \$(1...

7. Two random variables X and Y have joint probability density function s(x, y) = \$(1 – xy), 0<x< l; 0<y<l. The marginal pdfs for X and Y are respectively S(x) = {(2-x) 0<x< 1; s()= (2-y) 0<y<l. Determine the conditional expectation E(Y|X = x) and hence determine E(Y) [7] (ii) [3] Verify your answer to part (i) by calculating the value of E(Y) directly from the marginal pdf for Y. [Total 10]

• ### The joint PDF of random variables X and Y is expressed as (a) Determine the constant...

The joint PDF of random variables X and Y is expressed as (a) Determine the constant c. (b) Determine the marginal density function for X. (c) Determine the marginal density function for Y. (d) Are X and Y statistically independent? (e) Determine the probability of P(X ≤ 0.5 | Y = 1). The joint PDF of random variables X and Y is expressed as certy, 05xs1 and 05ys2 fx.x(x, y) = 10. elsewhere. (a) Determine the constant c. (b) Determine...

• ### Let the random variables X, Y with joint probability density function (pdf) fxy(z, y) = cry,...

Let the random variables X, Y with joint probability density function (pdf) fxy(z, y) = cry, where 0 < y < z < 2. (a) Find the value of c that makes fx.y (a, y) a valid pdf. (b) Calculate the marginal density functions for X and Y (c) Find the conditional density function of Y X (d) Calculate E(X) and EYIX) (e Show whether X. Y are independent or not.

• ### Consider the joint PDF of two random variables X and Y below. fx.y (x y) =...

Consider the joint PDF of two random variables X and Y below. fx.y (x y) = 1, if 0 < x < 1, and 0 y< 1, and fxx (г, у) Oif andy are outside of that square. So, basically, the joint PDF is a constant over the unit square Let W X+Y. Suppose we express the CDF of W in the usual double integral form h Fw(W) 2 dy dx g where w-0.4 is a given value at which...

• ### 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:

• ### Two random variables have joint PDF of F(x, y) = 0 for x < 0 and...

Two random variables have joint PDF of F(x, y) = 0 for x < 0 and y < 0 for 0 <x< 1 and 0 <y<1 1. for x > 1 and y> 1 a) Find the joint and marginal pdfs. b) Use F(x, y) and find P(X<0.75, Y> 0.25), P(X<0.75, Y = 0.25), P(X<0.25)

• ### 7. Suppose the random variables (X, Y) have joint pdf given by Ta ſ cx2y2 if...

7. Suppose the random variables (X, Y) have joint pdf given by Ta ſ cx2y2 if 0 <x sys1, l o otherwise. a. Find the constant c. b. Find the marginal density of X. c. Find the marginal density of Y. d. Are X and Y independent?

• ### 2. Let the random variables X and Y have the joint PDF given below: (a) Find P(X + Y ≤ 2). (b) Find the marginal PDFs o...

2. Let the random variables X and Y have the joint PDF given below: (a) Find P(X + Y ≤ 2). (b) Find the marginal PDFs of X and Y. (c) Find the conditional PDF of Y |X = x. (d) Find P(Y < 3|X = 1). Let the random variables X and Y have the joint PDF given below: 2e -0 < y < 00 xY(,) otherwise 0 (a) Find P(XY < 2) (b) Find the marginal PDFs of...