Section 4.3: The Normal Distribution

1. State the probability density function for a Gaussian random variable with parameters \(\mu\) and \(\sigma^{2}\).
2. Recognize the notation \(X \sim N(\mu, \sigma^{2})\) as indicating that \(X\) is a Gaussian random variable with parameters \(\mu\) and \(\sigma^{2}\).
3. State the mean and variance of a Gaussian random variable with parameters \(\mu\) and \(\sigma^{2}\).
4. Sketch a graph of the probability density function for a Gaussian random variable with parameters \(\mu\) and \(\sigma^{2}\), getting the general shape and placement of the probability density function correct, and use this graph to reason about the area under consideration for a given probability query.
5. Sketch a graph of the cumulative distribution function for a Gaussian random variable with parameters \(\mu\) and \(\sigma^{2}\), getting the general shape and placement of the cumulative distribution function correct.
6. State the definition of a standard Gaussian random variable, and recognize the notation that \(Z\) will often be used to denote a standard Gaussian random variable.
7. Recognize and use the convention of denoting the cumulative distribution function for a standard Gaussian random variable via \(\Phi(z) = P(Z \leq z)\).
8. Standardize a random variable \(X \sim N(\mu, \sigma^{2})\), and indicate the distribution of the transformed random variable.
9. Compute probability queries for a Gaussian random variable using both a table of standard normal probabilities via standardizing \(X\) and R.
10. Define the \(p\)-th percentile of a Gaussian random variable, and determine the \(p\)-th percentile using both a table of standard normal probabilities and R.