3.17 A PDF for a continuous random varaiable X is defined by C 0<x<2 2C4<< 6...
1. Let X be a continuous random variable with support (0, 1) and PDF defined by f(x) = ( cxn 0 < x < 1 0 otherwise, for some n > 1. a) Find c in terms of n. b) Derive the CDF FX(x).
2. A continuous random variable X has PDF SPI? 1€ (-2,2] fx() = 0 otherwise (a) Find the CDF Fx (x). (b) Suppose 2 =9(X), where gle) = { " Find the (DF, PDF of
Let X be a continuous random variable with PDF fx(x)- 0 otherwise We know that given Xx, the random variable Y is uniformly distributed on [-x,x. 1. Find the joint PDF fx(x, y) 2. Find fyo). 3. Find P(IYI <x3) Let X be a continuous random variable with PDF fx(x)- 0 otherwise We know that given Xx, the random variable Y is uniformly distributed on [-x,x. 1. Find the joint PDF fx(x, y) 2. Find fyo). 3. Find P(IYI
4. et X be a continuous random variable with support (0, 2) and PDF defined by f(x) = ( cx3 0 < x < 2 0 otherwise. a) Compute E[X]. b) Compute V ar(X)
LI CONTINUOUS DIST Let X be a random variable with pdf -cx, -2<x<0 f(x)={cx, 0<x<2 otherwise where c is a constant. a. Find the value of c. b. Find the mean of X. C. Find the variance of X. d. Find P(-1 < X < 2). e. Find P(X>1/2). f. Find the third quartile.
4. (20%) Let X be a continuous random variable with the following PDF Sce-4x 0<x fx(x) = to else where c is a positive constant. (a) (5%) Find c. (b) (5%) Find the CDF of X, Fx(x). (c) (5%) Find Prob{2<x<5} (d)(5%) Find E[X], and Var(X).
5. (50pt) X and Y are continuous random variables with pdf f(x, y) 2r for 0 < x y < 1, and f(x,y) = 0 otherwise. Find the conditional expectation of Y given X = z.
The random variables X and Y have joint PDF fX,Y(x,y) = {12x2y 0<=x<=c; 0 <= y <= 3 { 0 otherwise (a) FInd the value of C (b) Find the PDF fW(w) where W = X / Y (c) Find the PDF fZ(z) where Z = min(X,Y)
pectively, 3. Ajoint pdf is defined by (C(x + 2y), for 0 <x< 2, and 0 < y< 1, fx.r(x,y) = -{-4** 0. otherwise. a. Find the value of C. b. Find the marginal pdf of X alone. 9 c. Find the pdf of U = U = x+132 4. Consider n independent rvs XX2, ..., X, having the same distribution with a common variance a?. For any i = 1,2, ..., n, find Cov(x,- 8, 8), where 8 =...
Question 3 [17 marks] The random variables X and Y are continuous, with joint pdf 0 y otherwise ce fxx (,y) a) Show that cye fr (y) otherwise and hence that c = 1. What is this pdf called? (b) Compute E (Y) and var Y; (c) Show that { > 0 fx (a) e otherwise (d) Are X and Y independent? Give reasons; (e) Show that 1 E(XIY 2 and hence show that E (XY) =. Question 3 [17...