Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer n 2 2 such that Un > Un-1. Show that for each real number 0&...
Let Ui and U2 be independent random variables, each one distributed uniformly on Z be the minimum, Z = min{U1, U2} and W be the maximum, W = max{U1, U2}. Find the joint p.d.f of Z and W [0, 1]. Let Let Ui and U2 be independent random variables, each one distributed uniformly on Z be the minimum, Z = min{U1, U2} and W be the maximum, W = max{U1, U2}. Find the joint p.d.f of Z and W [0,...
1 (10pts) Let U1, U2, ... ,Un be independent uniform random variables over [0, 0] with the probability density function (p.d.f). () = a 2 + [0, 03, 0 > 0. Let U(1), U(2), .-. ,U(n) be the order statistics. Also let X = U(1)/U(n) and Y = U(n)- (a) (5pts) Find the joint probability density function of (X, Y). (b) (5pts) From part (a), show that X and Y are independent variables.
3. If U1 and U2 are independent standard uniform random variables, show that the variables are independent and identically distributed from N(0, 1) (the standard normal distribution) [10 marks
6. Let Ui, U,Un be independent Unif-2,0) random variables and Xa)in(U, U2, .., Un). Prove that X(a) converges in probability to -2
3. Let U1, U2,. be a sequence of independent Ber(p) random variables. Define Xo 0 and Xn+1-Xn +2Un-1, 1,2,.. (a) Show that X, n 0,1,2, is a Markov chain, and give its transition graph. (b) Find EX and Var(X) c)Give P(X
(1 point) Let A, B, and C be independent random variables, uniformly distributed over [0,4], [O,7], and [0, 6] respectively. What is the probability that both roots of the equation Ax2 Bx+ C = 0 are real? (1 point) Let A, B, and C be independent random variables, uniformly distributed over [0,4], [O,7], and [0, 6] respectively. What is the probability that both roots of the equation Ax2 Bx+ C = 0 are real?
Let X and Y be independent random variables uniformly distributed on the interval [1,2]. What is the moment generating function of X + 2Y? Let X and Y be independent random variables uniformly distributed on the interval [1,2]. What is the moment generating function of X + 2Y?
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent. Unif (0, 1) 5. Suppose U1 and...
(10 points) Consider the infinite sequence of independent and identically distributed (stan- dard) uniform random variables: U1, U2, ..., i.e., Ui » Uniform(0,1). Also let N ~ Poisson(a). Assume N is independent of {U;}i>1. Consider the random variable z = į V. Calculate EZ. (Hint: use conditioning.)
Let Y_(1) and Y_(2) be independent and uniformly distributed random variables over the interval (0,1). Find P(2 Y_(1)<Y_(2)).