Let Z ~ N(0, 1) and X ~ (µ,o²). This means that Z is a standard normal random variable with mean 0 and variance 1, while X is a normal random variable with mean u and variance o?. (a) Calculate E(Z³) (this is the \emph{third moment} of Z). (b) Calculate E(X³). Hint: Do not integrate the density function of $X$ unless you love messy integration. Instead use the fact that X can be represented as X = oZ + µ and expand the cube inside the expectation
Let Z ~ N(0, 1) and X ~ (µ,o²). This means that Z is a standard normal random variable with mean 0 and variance 1, while X is a normal random variable with mean u and variance o?. (a) Calculate E(Z³) (this is the \emph{third moment} of Z). (b) Calculate E(X³). Hint: Do not integrate the density function of $X$ unless you love messy integration. Instead use the fact that X can be represented as X = oZ + µ and expand the cube inside the expectation
A First Course in Probability (10th Edition)
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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Transcribed Image Text:Let \( Z \sim \mathcal{N}(0,1) \) and \( X \sim \mathcal{N}(\mu, \sigma^2) \). This means that \( Z \) is a standard normal random variable with mean 0 and variance 1, while \( X \) is a normal random variable with mean \( \mu \) and variance \( \sigma^2 \).
(a) Calculate \( \mathbb{E}(Z^3) \) (this is the third moment of \( Z \)).
(b) Calculate \( \mathbb{E}(X^3) \).
**Hint:** Do not integrate the density function of \( X \) unless you love messy integration. Instead, use the fact that \( X \) can be represented as \( X = \sigma Z + \mu \) and expand the cube inside the expectation.
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