Problem 4. Let X and Y be independent Geom(p) random variables. Let V - min(X, Y)...
Problem 41.3 Let X and Y be independent random variables each geometrically distributed with parameter p, i.e. p(1- p otherwise. Find the probability mass function of X +Y
2. Let U and V be independent random variables, with P(U 1) 1/4 and P(U = -1) = P(V -1) 1) = P(V 3/4. Define X = U/V and Y = U V (a) Give the joint pmf of X and Y [4] (b) Calculate Cov(X,Y) [4] 2. Let U and V be independent random variables, with P(U 1) 1/4 and P(U = -1) = P(V -1) 1) = P(V 3/4. Define X = U/V and Y = U V...
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
5. (a) (6 marks) Let X be a random variable following N(2.4). Let Y be a random variable following N(1.8). Assume X and Y are independent. Let W-min(x.Y). Find P(W 3) (b) (8 marks) The continuous random variables X and Y have the following joint probability density function: 4x 0, otherwise Find the joint probability density function of U and V where U-X+Y and -ky Also draw the support of the joint probability density function of Uand V (o (5...
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random variables Z and W (b) Find the density of random variable W (c) Find the density of random variable Z The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random...
Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x, 0<x<1 (zero otherwise), and g(y) = 3y2, 0<y<1 (zero otherwise). Let U = min(X,Y), and V = max(X,Y). Find the joint pdf of U and V.
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b) 3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
Problem 8. Suppose that XGeom(p) and Y ~ Geom(r) are independent. Find the probability P(X <Y).
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3) and Y is binomial(n = 3,p = 1/3). Find the moment-generating functions of the three random variables X, Y and X + Y . (You may look up the first two. The third follows from the first two and the behavior of moment-generating functions.) Now use the moment-generating function of X + Y to find the distribution of X + Y .
Let X and Y be independent exponential random variables with pdfs f(x) = λe-λx (x > 0) and f(y) = µe-µy (y > 0) respectively. (i) Let Z = min(X, Y ). Find f(z), E(Z), and Var(Z). (ii) Let W = max(X, Y ). Find f(w) (it is not an exponential pdf). (iii) Find E(W) (there are two methods - one does not require further integration). (iv) Find Cov(Z,W). (v) Find Var(W).