The t-distribution is used in place of the standard normal distribution for small sample sizes ($n<30$). Let $Z \sim N(0,1)$ and $W \sim \chi^2(n)$ be independent random variables. The t-distribution is given by $T = \frac{Z}{\sqrt{W/n}}$ As the degrees of freedom $n$ gets larger, the t-distribution approximates the normal distribution. #expand Week 3 Lesson 3 DS5003 ## Notation $T \sim t(n)$ with $n$ [[degrees of freedom]]. ## Probability density function The pdf of the t-distribution can be derived, however it is almost never used in closed form and instead we lookup values in [[R]]. ## Expectation $E(X) = 0$ ## Variance $V(X) = \frac{n}{n - 2}$ ## R The t-distribution is `t` in [[R]]. ```R # Quantile function (to get confidence interval) n <- 20 df <- n - 1 qt(0.975, df) ```