A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Question
1. Consider the function \( f(x) = \frac{(x+1)^2}{15} \) for \( x = -2, -1, 0, 1, 2 \). a) Verify that the function is a pdf. b) Find \( E[X] \) directly, that is, evaluate \(\sum [x \cdot f(x)]\). c) Find the moment generating function for \( X \). d) Use the moment generating function to find \( E[X] \), verifying your answer to part b). e) Find \( E[X^2] \) directly. f) Use the moment generating function to find \( E[X^2] \). g) Find the variance of \( X \) and the standard deviation of \( X \).
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Transcribed Image Text:1. Consider the function \( f(x) = \frac{(x+1)^2}{15} \) for \( x = -2, -1, 0, 1, 2 \). a) Verify that the function is a pdf. b) Find \( E[X] \) directly, that is, evaluate \(\sum [x \cdot f(x)]\). c) Find the moment generating function for \( X \). d) Use the moment generating function to find \( E[X] \), verifying your answer to part b). e) Find \( E[X^2] \) directly. f) Use the moment generating function to find \( E[X^2] \). g) Find the variance of \( X \) and the standard deviation of \( X \).
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