1.a) Find the conditional probability of a variable Y given X=x for a joint distribution with a function f(x,y)=xe-xy for y>0     b) Then calculate the conditional probability for Y>0.5 when X=0.3   2. a)Determine the value c of the joint distribution with a density function f(x,y)=c(x-1)y for 0<x<4 and 0<y<x.     b)For the same function calculate the probabilities of X<2 and Y<2.

Suppose that X and Y have a continuous joint distribution for which the joint probability den- sity function is defined as follows: fx,y (x, y) = { vº $4² for 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1 otherwise. (a) Determine P(X 1/2). (b) Determine the marginal probability density functions of X and Y. (c) Are X and Y independent? Explain.

5. Let X have a uniform distribution on the interval (0,2). Given X = x, let Y have a (conditional) uniform distribution on the interval (0,2x). (a) Write down the conditional probability density function of Y given that X = x. Make sure to include the support of Y for each possible value of x. (b) Find E(Y|x) and E(Y). (c) Find Var(Y|æ) and Var(Y). (d) Determine f(x,y), the joit probability density function of (X,Y).

Suppose that X and Y have a joint probability density function given by

Let X and Y are two random variables with p.d.f as F(x,y) = 3x^2y+3xy^2,0 < x< 1,0 < y < 1. (i) Find conditional density of X given Y. (ii) Find the mean, mode and standard deviation of r.v Y. (iii) Correlation between X and Y. (iv) Check that X and Y are independent or not?

2. Let X be a Chi-squared random variable with density function fx(x) = exp (-;), a x > 0. Suppose Y is independent of, and has the same distribution as, X. (a) Let Z = VX. Show that the density function of Z is given by Sale) = 2a exp (-) z > 0. Hence state the value of a.

Let X and Y have the joint probability density functionf(x, y) = 5, for 0 < x < y < 1.(i) Find the marginal probability density function of X and Y.(ii) Find the conditional probability density function of X, given Y = y.(iii) Find the conditional mean of X, given Y = y.(iv) Find the conditional variance of X, given Y = y.

Suppose that X and Y have a joint probability density function given by ce-3z=5y if a, y 20 fx.y(T, y) = otherwise Find the marginal probability density function fx and state the name of the distribution of X.

2. The continuous random variables X and Y have known joint probability density function 4x fxy(x,y) given by f(x, y) = y +1 0≤ y ≤x≤1 0 a) Determine the covariance otherwise cov(X,Y) of the random variables X and Y. b) Determine the correlation coefficient Pxx of the random variables X and Y. c) Determine the expected value E{g(X,Y)} for the case where g(x, y) = x² + y². d) Determine the variance var{X+Y}.