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 R.
10. Define the $p$-th percentile ($q$ quantile) of a Gaussian random variable, and determine the $p$-th percentile ($q$ quantile) using R.