Question

2. Let X be a continuous random variable with pdf ( cx?, |a| 51, f(x) = { 10, otherwise, where the parameter c is constant (with respect to x). (a) Find the constant c. (b) Compute the cumulative distribution function F(x) of X. (c) Use F(x) (from b) to determine P(X > 1/2). (d) Find E(X) and V(X).

Answer 1

TOPIC:Pdf,cdf,probability,expectation and variance.

27 The pdf of x is given as - Hx) = { cx² ; 1x141. Ey / sxsl. otherwise. @ Dince, HX) is a valid py; we, must have, Haldx=1. > c. (x²dx=1. 6 5 LP-C33 So, the pdf of x reduces to Hex) = { { x :-s** i otherwise.

by Define F. (x) = P (XLX). . =cdf of x. So, we have for, -15X61; F(x) = (det) dt . = ² . [2² - (-2²). - Xt; dom, -15. We get, x F(x) = { 0 ; x<-1 x + -I 251. 2 1 : x²l. Ans,

③ we needs P(xy, ť). = 1-P ( x < Ž). - 1-F (2 [F(x) = P(x <x): as, x is continuous = 1 - ² - [ (4) + 1] (using the ed - 0.4375 anys - we need, E(X). afbrydr. - L-. nom.

Again, we have € (x²) - 1 x pex) dx *[T. x2. Hence, var (x) = E(X²) – 6²(x). - -(0) Tons