Chapter 43: Moment Generating Functions Part 2

Recap: What is an mgf?

Example 1.   Let \(X\) be a random variable with mgf \[M_X(t)= \frac{1}{5}e^t + \frac{3}{10}e^{2t} + \frac{1}{2}e^{3t}.\] Find the pmf or pdf of \(X\).

Example 2.   Let \(X\) be a normal random variable with mean \(\mu\) and variance \(\sigma^2\), i.e. \(X \sim N(\mu,\sigma^2)\).

  1. Find the mgf of \(X\).

  2. Find \(\mathbb{E}[X]\).

  3. Find \(Var(X)\).

Theorem 3.   Let \(X\) have mgf \(M_X(t)\), and let \(Y=aX+b\), where \(a\) and \(b\) are constants. Then \[M_Y(t)=\]

Proof. Proof. ◻

Question: Do linear transformations always preserve the distribution type?

I.e., if \(X\) has a certain probability distribution, does \(aX+b\) always have the same distribution type?

Example 4.   Let \(X \sim U[0,1]\), and \(Y = 2X+3\). Is \(Y\) also a uniform rv? If so, what are its parameters?

Example 5.   Let \(X \sim Exp(\lambda=5)\), and \(Y = 2X+3\). Is \(Y\) also an exponential rv? If so, what is its parameter?

Mgf’s of Sums of Independent rv’s

Theorem 6.   Let \(X_1, X_2, \ldots, X_n\) be independent rv’s with respective mgf’s \(M_{X_i}(t)\), for \(i=1,2,\ldots,n\). Let \(Y=\sum_{i=1}^n a_iX_i\), where \(a_i\) are constants. Then \[M_Y(t)= %\Pi_{i=1}^n M_{X_i}(a_it).\]

Proof. Proof. ◻

Example 7.   Let \(X_i \sim N(\mu_i, \sigma_i^2)\) be independent normal rv’s. What is the distribution of  \(Y=\sum_{i=1}^n X_i\)?

Example 8.   Let \(X_i \sim N(\mu, \sigma^2)\) be iid normal rv’s, for \(i=1,2,\ldots,n\). What is the distribution of  \(\bar{X}=\frac{\sum_{i=1}^n X_i}{n}\)?

Example 9.   Let \(Z\) be a standard normal random variable, i.e. \(Z \sim N(0,1)\). Show that \(Z^2 \sim \chi_1^2\), i.e. is a chi-squared rv with 1 degree of freedom.