Let X be a RV with PMF Px(x) = x²/c |0 x = −3, −2, −1, 0, 1, 2, 3, otherwise. (a) Find c and E[X]. (b) What is the PMF of the random variable Z = (X – E[X])²? (c) Using the results from (b), find the variance of X. (d) Find the variance of X using the formula var(X) = Σx(x − E[X])²px(x).
Let X be a RV with PMF Px(x) = x²/c |0 x = −3, −2, −1, 0, 1, 2, 3, otherwise. (a) Find c and E[X]. (b) What is the PMF of the random variable Z = (X – E[X])²? (c) Using the results from (b), find the variance of X. (d) Find the variance of X using the formula var(X) = Σx(x − E[X])²px(x).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![1. Let \( X \) be a random variable (RV) with probability mass function (PMF):
\[ P_X(x) =
\begin{cases}
\frac{x^2}{c} & \text{if } x = -3, -2, -1, 0, 1, 2, 3, \\
0 & \text{otherwise.}
\end{cases}
\]
Tasks:
(a) Find \( c \) and \( E[X] \).
(b) What is the PMF of the random variable \( Z = (X - E[X])^2 \)?
(c) Using the results from (b), find the variance of \( X \).
(d) Find the variance of \( X \) using the formula \(\text{var}(X) = \sum_x (x - E[X])^2 p_X(x)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F230dd6fc-67da-4f36-8738-d9d162992241%2Fdbd9fd71-44c5-4448-81ca-979859da07cd%2Fbdhtb3c_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let \( X \) be a random variable (RV) with probability mass function (PMF):
\[ P_X(x) =
\begin{cases}
\frac{x^2}{c} & \text{if } x = -3, -2, -1, 0, 1, 2, 3, \\
0 & \text{otherwise.}
\end{cases}
\]
Tasks:
(a) Find \( c \) and \( E[X] \).
(b) What is the PMF of the random variable \( Z = (X - E[X])^2 \)?
(c) Using the results from (b), find the variance of \( X \).
(d) Find the variance of \( X \) using the formula \(\text{var}(X) = \sum_x (x - E[X])^2 p_X(x)\).
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