410Hw02ans.pdf - STAT 410 Homework#2(due Friday February 8 by 4:00 p.m Spring 2019 A Stepanov Please include your name with your last name underlined

# 410Hw02ans.pdf - STAT 410 Homework#2(due Friday February 8...

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STAT 410 Homework #2 (due Friday, February 8, by 4:00 p.m.) Spring 2019 A. Stepanov Please include your name ( with your last name underlined ) , your NetID, and your section number at the top of the first page. No credit will be given without supporting work. If the answer is a function, its support must be included. 1. Consider a mixed random variable X with the cumulative distribution function F ( x ) = 2 0 0.3 0.3 0.6 0.6 0.9 1 0.9 x x x x x x a) Identify the discrete portion of the probability distribution. b) Identify the continuous portion of the probability distribution. c) Find E ( X ) .
a) F ( x ) “jumps” at x = 0.3 from 0.00 to 0.09, size of the “jump” = 0.09 – 0.00 = 0.09, at x = 0.6 from 0.36 to 0.60, size of the “jump” = 0.60 – 0.36 = 0.24, at x = 0.9 from 0.90 to 1.00, size of the “jump” = 1.00 – 0.90 = 0.10. Discrete portion of the probability distribution of X: p ( 0.3 ) = 0.09, p ( 0.6 ) = 0.24, p ( 0.9 ) = 0.10. b) Continuous portion of the probability distribution of X: f ( x ) = F ' ( x ) = 2 0.3 0.6 1 0.6 0.9 0 otherwise x x x . c) E ( X ) = 0.3 0.09 + 0.6 0.24 + 0.9 0.10 + 0.6 0.3 2 x x dx + 0.9 0.6 1 x dx = 0.027 + 0.144 + 0.090 + 3 0.6 0.3 2 3 x + 2 0.9 0.6 2 x = 0.261 + 0.126 + 0.225 = 0.612 . OR Since X is a nonnegative random variable, E ( X ) = 0 1 F x dx = 0.3 0 1 dx + 0.6 2 0.3 1 x dx + 0.9 0.6 1 x dx = 0.3 0 x + 3 0.6 0.3 3 x x

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