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 \( X \).

Answered: a) Verify that the function is a pdf.…

a) Verify that the function is a pdf. b) Find E[X] directly, that is evaluate sum[x*f(x)] c) Find the moment generating function for X.

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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