Chapter 6: Some Continuous Distributions

1. State the probability density function for a general Gaussian (normal) random variable with mean \(\mu\) and variance \(\sigma^{2}\).
2. Compute any probability query \(P(a < X < b)\) for a Gaussian random variable using the cumulative distribution function \(\Phi(z)\) for a standard Gaussian random variable using only positive values of \(z\).
3. Determine the approximate \(p\)-th percentile for a \(N(\mu, \sigma^{2})\) random variable using the cumulative distribution function \(\Phi(z)\) for a standard Gaussian random variable using only positive values of \(z\).
4. State the assumptions and conclusion of the Central Limit Theorem.
5. Use the Central Limit Theorem to approximate probabilities for sample means and totals of a random sample \(X_{1}, X_{2}, \ldots, X_{n} \stackrel{\text{iid}}{\sim} D(\mu, \sigma^{2})\).
6. Recognize that Gaussian random variables are closed under linear combination (i.e. the linear combination of Gaussian random variables is itself Gaussian), and use this result in the special case where the Gaussian random variables are independent.
7. Use the Gaussian approximation for the sampling distribution of the sample mean to approximate probabilities about a binomial random variable, both with and without a continuity correction.
8. State the moment generating function of a Gaussian random variable.
9. Define the transformation of a Gaussian random variable that gives rise to a log-normal random variable.
10. Relate the moment generating function of a Gaussian random variable to the mean and variance of a log-normal random variable.
11. Give a constructive definition of a chi-squared random variable.
12. Relate the distribution of a chi-squared random variable to the distribution of a Gamma random variable.