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