Problem 5) Let X and Y be independent gamma RVs with parameter (a, 1) and (3,...
Problem 1. (5 marks. 3. 2) Assume X ~ Gamma(01, β) and Y ~ Gamma(O2, β) are independent random variables. a) Compute the Joint density of U = X + Y and V X X + Y , be sure to include the support/domain. b) Based on the joint density derived in part (a) find the marginal densities of U and V, be sure to include the support (s)/domain(s). Explicitly state the name of the distributions of U and V...
Let X and Y be independent exponential(1) RVs (f(x) e 10). Show that uniform(0, 1) distribution. Hint: consider defining the auxiliary X/(X Y) has a RV XY [12
Let X and Y be independent exponential random variables with parameter 1. Find the joint PDF of U and V. U = X + Y and V = X/(X + Y)
2. Let X and Y be independent integer-valued RVs with given distributions jezqj = 1. (a) Compute the probability P(X = IYI). (b) Compute the probability P(Y/X є Z). 2. Let X and Y be independent integer-valued RVs with given distributions jezqj = 1. (a) Compute the probability P(X = IYI). (b) Compute the probability P(Y/X є Z).
Exercise 6.17. Let U and V be independent, U~ Unif(0,1), and V~ Gamma(2.A) which means that V has density function fv(1) λ2e-W for v0 and zero elsewhere. Find the joint density function of (X, Y)-. (UV, ( 1-U)V). Identify the joint distribution of (X, Y) In terms of named distributions. This exercise and Example 6.44 are special cases of
(a) Let x, have a chi-squared distribution with parameter y, and let x, be independent of x, and have a chi-squared distribution with parameter vz Show that x2 + x, has a chi-squared distribution with parameter v, + vz Let y = x2 + x2. Identify the correct expression for Fyn). "1/2 - 2x212/2-12-(x1 + x2)/2 dx1 OFyly) = ' -X1/2 dx1 + -x212 dxz e 109) = 6 {1*(***) ** (*)*" + x2)120mg +6 (277(3)) -<*****@*)*** .(-)70%as C (7-(1)...
7. Assume X gammala, y) and Y-gamma(8,7) are independent. (a) Show that U = X + Y gamma(a +.). (b) Show that V = X/(X+Y) beta(a. 8). (c) Show that U and V are independent. (d) Show that W = 72 gamma(1/2, 1/2) if Z N (0,1).
Let X1,X2 be two independent exponential random variables with λ=1, compute the P(X1+X2<t) using the joint density function. And let Z be gamma random variable with parameters (2,1). Compute the probability that P(Z < t). And what you can find by comparing P(X1+X2<t) and P(Z < t)? And compare P(X1+X2+X3<t) Xi iid (independent and identically distributed) ~Exp(1) and P(Z < t) Z~Gamma(3,1) (You don’t have to compute) (Hint: You can use the fact that Γ(2)=1, Γ(3)=2) Problem 2[10 points] Let...
Problem B. Let X and Y have joint density Show that Y and X/Y are independent.
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...