# Week 9

This week, we will discuss functions of random variables and sampling distributions.
Published

March 27, 2023

## Learning Outcomes

### First Lecture

• Functions of Random Variables

### Second Lecture

• Sampling Distributions

## Important Concepts

### First Lecture

#### Functions of Random Variables

##### Method of Distribution Function

Let $$Y=g(X)$$ be a function of a Random variable $$X$$. The density function of $$Y$$ can be found with the following steps:

1. Find the region of $$Y$$ in the $$X$$ space.
2. Find the region of $$Y\leq y$$
3. Find $$F_Y(y)=P(Y\le y)$$ by integrating over the region of $$Y\le y$$
4. Find the density function $$f_Y(y)=\frac{dF_Y(y)}{dy}$$
##### Method of Transformations

Let $$Y=g(X)$$ be a function of a Random variable $$X$$, if $$g(X)$$ is either increasing or decreasing for all values of $$X$$ such that $$f_X(x)>0$$. Then the density function of $$Y=g(X)$$ is given as

$f_Y(y) = f_X\{g^{-1}(y)\}\left|\frac{dh^{-1}(y)}{dy}\right|$

##### Method of Moment Generating Functions

Let $$Y=g(X)$$ be a function of a Random variable $$X$$. Find the moment generating function $$M_Y(t)$$. Compare $$M_Y(t)$$ to known moment generating functions.

### Second Lecture

#### Sampling Distributions

##### Observing Random Variables

When collecting a sample of $$n$$, we tend to observe individual random variables: $$\{X_1, X_2, \cdots,X_n\}$$.

##### Sum of Random Variables

Let $$X_i$$, for $$i=1, \cdots, n$$, be identically and independently distributed (iid) normal distribution with mean $$\mu$$ and variance $$\sigma^2$$. Let $$T=\sum_{i=1}^nX_i$$ follow an normal distribution with mean $$\mu$$ and variance $$n\sigma^2$$.

#### Central Limit Theorem

Let $$X_1, X_2, \ldots, X_n$$ be identical and independent distributed random variables with $$E(X_i)=\mu$$ and $$Var(X_i) = \sigma²$$. We define

$Y_n = \sqrt n \left(\frac{\bar X-\mu}{\sigma}\right) \mathrm{ where }\ \bar X = \frac{1}{n}\sum^n_{i=1}X_i.$

Then, the distribution of the function $$Y_n$$ converges to a standard normal distribution function as $$n\rightarrow \infty$$.

#### Other Sampling Distributions

##### $$\chi^2$$-distribution

Let $$Z_1, Z_2,\ldots,Z_n \overset{iid}{\sim}N(0,1)$$,

$\sum_{i=1}^nZ_i^2\sim\chi^2_n.$

Let $$X_1, X_2,\ldots,X_n \overset{iid}{\sim}N(\mu,\sigma^2)$$, $$S^2 = \frac{1}{n-1}\sum^n_{i=1}(X_i-\bar X)^2$$, and $$\bar X \perp S^2$$; therefore:

$\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}.$

##### t-distribution

Let $$Z\sim N(0,1)$$, $$W\sim \chi^2_\nu$$, $$Z\perp W$$; therefore:

$T=\frac{Z}{\sqrt{W/\nu}} \sim t_\nu$

##### F-distribution

Let $$W_1\sim\chi^2_{\nu_1}$$ $$W_2\sim\chi^2_{\nu_2}$$, and $$W_1\perp W_2$$; therefore:

$F = \frac{W_1/\nu_1}{W_2/\nu_2}\sim F_{\nu_1,\nu_2}$

## Resources

### First Lecture

Slides Videos Notes
Slides Video 001 Video 002 Notes

### Second Lecture

Slides Videos Notes
Slides Video 001 Video 002 Notes