8. Let X.(i-12) be independent N(0,1) random variables. a. Find the value of c such that...
8. Let X (i-1,2) be independent N(0,1) random variables. a. Find the value of c such that P ( (X1 + X2 )2/( X2 -X1)2 < c ) =.90 b. Find P(2 X1 -3 X2< 1.5) c. Find 95th percentile of the distribution of Y-2 X1 -3 X2
Let X1 and X2 be independent n(0,1) random variables. Find the pdf of (X1 - X2)^2/2
3. Let Z be a continuous random variable with Z-N(0,1). (a) Find the value of P(Z <-0.47). (b) Find the value of P(Z < 2.00). Note denotes the absolute value function. (c) Find b such that P(Z > b) = 0.9382. (d) Find the 27th percentile. (e) Find the value of the critical value 20.05-
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3). 3. (25 pts.) Let X1,...
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y = X1 + X, and Z X2 + X3 a.) Compute E[X, X,X3]. (5 points) b.) Compute Var(x1). (5 points) c.) Compute and draw a graph of the density function fy (15 points)
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
5. Let X; (i = 1, 2, 3) be be independent gamma random variables with a; = i and B. = 8. a. Find a maximum likelihood estimator of 8 and prove that it is unbiased. b. Show that 2(X1+X2+Xa) is a pivotal quantity for 0. c. Find a 95% confidence interval for 6.
Problem 8. Let X1, X2, , Xn be independent ฆ(0,1) random variables. Let m,-1 for k 1,2,3. Are there numbers mi,m2, m3 such that n.y rn1 m1 a.S n3 m3 holds? If so, calculate these numbers.
2. (20 pts.) Let X1,.., X45 be i.i.d. Uniform[0,1] random variables. Find (approximately) the probability P[12 X3++Xx18 2. (20 pts.) Let X1,.., X45 be i.i.d. Uniform[0,1] random variables. Find (approximately) the probability P[12 X3++Xx18
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .