(35) Let X and Y be discrete random variables with join mass function 14 p(x, y) = (a) Find the marginal mass functions of X and Y, fx and fy, respec- tively. (b) Find the constant k (c) Find Cov(X,...
8. Let X be a discrete random variable with the mass function fx (0 ) = 1-p and fx(1) = p. Let Y = 1 _ X and Z = XY. (a) Find the joint mass functions of X and Y. (b) Find the joint mass functions of X and Z
sity functions. Exercise 6.46. Let X, Y be independent random variables with density functions fx and fy. Let T- min(X, Y) and V max(X, Y). Use the joint density function fr v from equation (6.31) to compute the marginal density functions fr of T and fv of V
2. Let X and Y be continuous random variables with joint probability density function fx,y(x,y) 0, otherwise (a) Compute the value of k that will make f(x, y) a legitimate joint probability density function. Use f(x.y) with that value of k as the joint probability density function of X, Y in parts (b),(c).(d),(e (b) Find the probability density functions of X and Y. (c) Find the expected values of X, Y and XY (d) Compute the covariance Cov(X,Y) of X...
Suppose two continuous random variables X and Y have cumulative distribution functions Fx(x) and Fy(y) respectively. Suppose that Fx(x) > Fy(x) for all x. Indicate whether the following statements are TRUE or FALSE with brief explanation. (a) E(X) > E(Y) (b) The probability density functions fx, fy satisfy fx(x) > fy(x) for all x. (c) P (X = 1) > P (Y = 1)
3. Consider two random variables X and Y, whose joint density function is given as follows. Let T be the triangle with vertices (0,0), (2,0), and (0,1). Then if (x, y for some constant K (a) (2 pts.) Find the constant K (b) (4 pts.) Find P(X +Y< 1) and P(X > Y). (c) (4 pts.) Find the marginal densities fx and fy. Conclude that X and Y are not independent
Problem 4 Let X be a discrete random variable with probability mass function fx(x), and let t be a function. Define Y = t(X): that is, Y is the randon variable obtained by applying the function t to the value of X Transforming a random variable in this way is frequently done in statistics. In what follows, let R(X) denote the possible values of X and let R(Y) denote the possible values of To compute E[Y], we could irst find...
(a) Find the constant c. (b) Find fX (x) and fY (y) (c)For0<x<1,findfY|X=x(y)andμY|X=x andσY2|X=x. (d) Find Cov(X, Y ). (e) Are X and Y independent? Explain why. 3. (50 pts) Let (X, Y) have joint pdf given by c, |y< x, 0 < x < 1, f(r,y)= 0, o.w., (a) Find the constant c (b) Find fx(x) and fy(y) and oyx (c) For 0 x 1, find fy\x= (y) and (d) Find Cov(X, Y) (e) Are X and Y independent?...
ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that the function p: R2 R defined by p(r, y) px(x)pr(y) is a joint PMF by verifying that it satisfies properties (a)-(c) of Proposition 6.1 on page 262. Hint: A subset of a countable set is countable CHAPTER SIX Joindy Discrete Random Variables 6.2 Joint and marginal PMFs of the discrete random variables x numher of bedrooms and momber of bwthrooms of a...
dont have to do part C! The join pdf of random variables X and Y is given as JXY, fxx(x, y) = {e=(x+y) x>0, else y>0 0 a) (10 pts) Find marginal pdf fx(x) for X, fy(y) for Y, and plot fx(x) and fy(y) b) (10 pts) Are X and Y independent? Why? c) (15 pts) Find the mean of X, the mean of Y, E[XY). d) (10 pts) Find the probability of event {Osxsys1}
. Let X and Y be two random variables with joint probability density function fx,y(x, y)-cy for 0 x 1 and 0 y 1. (Note: fxy(x,y) = 0 outside this domain ) (a) Find the marginal distribution fx(x). (b) Find the value of constant c, using the fact that fx,y(x, y) dx dy = 1.