91Ó°ÊÓ

Simple random variable \(X\) has distribution $$ X=\left[\begin{array}{lllll} -3 & -2 & 0 & 1 & 4 \end{array}\right] \quad P X=\left[\begin{array}{llllll} 0.15 & 0.20 & 0.30 & 0.25 & 0.10 \end{array}\right] $$ a. Determine the moment generating function for \(X\). b. Show by direct calculation the \(M_{X}^{\prime}(0)=E[X]\) and \(M_{X}^{\prime \prime}(0)=E\left[X^{2}\right]\).

Use the moment generating function to obtain the variances for the following distributions Exponential \((\lambda)\) Gamma \((\alpha, \lambda)\) Normal \(\left(\mu, \sigma^{2}\right)\)

Suppose the pair \(\\{X, Y\\}\) is independent, with both \(X\) and \(Y\) binomial. Use generating functions to show under what condition, if any, \(X+Y\) is binomial.

Use the central limit theorem to show that for large enough sample size (usually 20 or more), the sample average $$ A_{n}=\frac{1}{n} \sum_{i=1}^{n} X_{i} $$ is approximately \(N\left(\mu, \sigma^{2} / n\right)\) for any reasonable population distribution having mean value \(\mu\) and variance \(\sigma^{2}\)