Can a statistical model have two unknown parameters?

Question

Is an $N(\mu,\sigma^2)$ distribution a statistical model if both the parameters are unknown? The definition I have in front of me only refers to one unknown parameter.

Answer 1

Some distributions such as Poisson and exponential have only one parameter. Other distributions such as binomial and normal have two. (In the case of the binomial, the number of trials $n$ is often known, and the success probability $p$ is often unknown.) There are also distribution families with more than two parameters.

When you are doing inferential procedures (e.g., confidence intervals and tests of hypotheses) for data from a distribution with more than one parameter, it is important to understand from the start which parameters are unknown (and subject to inference) and which are known.

For example, suppose you have data $X_1, X_2, \dots, X_n$ from a normal population. If the mean $\mu$ is unknown and the standard deviation $\sigma = 3$ is known, then a 95% confidence interval for $\mu$ is $\bar X \pm 1.96 \sigma/\sqrt{n}.$ Thus if $n = 4$ and the sample mean is $\bar X = 3.25$ then the CI is $3.25 \pm 1.96(3/2)$ or $(0.31, 6.19)$.

By contrast, if $\sigma$ is unknown, and estimated by the sample standard deviation $S = 3.19$, then a 95% CI for $\mu$ is $\bar X \pm t^*S/\sqrt{4},$ where $t^* = 3.182$ is found from a table of Student's t distribution (with degrees of freedom $df = n - 1 = 3.$ Then the CI is computed as $3.25 \pm 3.182(3.19/2)$ or $(-1.825, 8.325).$ So knowing whether $\sigma$ is known or to be estimated by $S$ can make a noticeable difference in the CI.