5 and 6 please Prompt: Recall that XNu, o) if the PDF of X is given...
Suppose X is a continuous random variable having pdf (1+x, -1 < x < 0, f(x) = { 1 – x, 0 < x <1, lo, otherwise (a) Find E(X2). (b) Find Var(X2).
PROBLEM 3 Let X1, X2,L , X, be iid observations from a distribution with pdf given by f(xl0)=0x0-, 0<x<1, 0<O<00. a) Find the maximum likelihood estimator of O. b) Find the moment estimator of 0. c) (Extra credit) Compare the mean squared error of the two estimators in (a) and (b). Which one is better? (5 points)
1. Let X be a random variable with pdf f(x )-, 0 < x < 2- a) Find the cdf F(x) b) Find the mean ofX.v c) Find the variance of X. d) Find F (1.75) e) Find PG < x < +' f) Find P(X> 1). g) Find the 40th percentile.*
Example 46. Let X be a random variable with PDF liſa - 1), 1<a < 3; f(a) = { à(5 – a), 3 < x < 5; otherwise. Find the CDF of X. @ Bee Leng Lee 2020 (DO NOT DISTRIBUTE) Continuous Random Var Example 46 (cont'd). Find P(1.5 < X < 2.5) and P(X > 4).
6-x-4, 0x<2 0 1 2cych Exri If for two R.V. s X&Y the joint pdf is given by, otherwise Find Frix (o (1), Frix (alt), Ely/x-1]. var [Ylx-i] = E[^\x-]- (E[1\x=1])!
4. Suppose X and Y have the joint pdf f(x,y) = 6x, 0 < x < y < 1, and zero otherwise. (a) Find fx(x). (b) Find fy(y). (c) Find Corr(X,Y). (d) Find fy x(y|x). (e) Find E(Y|X). (f) Find Var(Y). (g) Find Var(E(Y|X)). (h) Find E (Var(Y|X)]. (i) Find the pdf of Y - X.
1 x Suppose X has an exponential distribution, thus its pdf is given by fx (x) = 5e8,0 5x<0, 2> 0;0 0.w. a. Find E(X) b. Find E(X(X-1) c. Find Var (x)
5. Y is a continuous random variable with pdf f(y) = (4 – y)/8, 0<y< 4. (a) Find E(Y). (b) Find E(Y2). (c) Find Var(Y).
Problem 2 (9 points) Consider a random variable X with pdf given by: (x) 0.06x +0.05 0x <5 4pts P (3.5 < X < 6.5)- Find Find Ex]- 5pts Problem 3 (9 points) Consider a random variable X with pdf given by: f(x) 0.06x +0.05 x -0, 1.5, 2, 4, 5 .ind P(3.5 <X <6.5)- 4pts 5pts Given that EN-3.46, find al
2. Let the pair (X,Y) have joint PDF fxy(x, y) = c, with 2.2 + y2 <1. (a) Find c and the marginal PDFs of X and Y. (b) What are the means of X and Y ? No calculations are needed, only a brief expla- nation is required. (c) Find the conditional PDF of Y given X = x and deduce E|Y|X = x]. (d) Obtain E(XY) and compare it to E[X]E[Y). (e) Are X and Y independent? Explain....