---
title: "Fitting Models"
author: "Prof. Eric A. Suess"
format: 
  html:
    self-contained: true
---

## Fitting Models

Random sample from the normal distribution, estimate the parameters

simulated sample

```{r}
mu <- 43; sigma <- 5.97   # parameters
n <- 300

x <- rnorm(n, mu, sigma)

hist(x, probability = TRUE)
```

Estimate the parameters with x.bar and sd

```{r}
mu.hat <- mean(x); mu.hat
sigma.hat <- sd(x); sigma.hat
```

Plot the fitted model on the histogram

```{r}
hist(x, probability = TRUE)
curve(dnorm(x, mu.hat, sigma.hat), add=T)
```

Random sample from the exponential distribution, estimate the parameters

simulated sample

```{r}
lambda <- 11.34   # parameter
n <- 30

x <- rexp(n, rate = lambda)

hist(x, probability = TRUE)
```

Estimate the parameters with x.bar

```{r}
lambda.hat = 1/mean(x); lambda.hat
```

Plot the fitted model on the histogram

```{r}
hist(x, probability = TRUE)
curve(dexp(x, rate = lambda.hat), add=T)
```

Now suppose we want to fit the gamma distribution to a random sample how can we do this?  

Can we use x.bar and the sd?

The big question is, how do we estimate parameters in models in general?