Week 4

This week, we will review functions of random variables and introduce the idea of sampling distributions and Central Limit Theorem.
Published

September 12, 2022

Learning Outcomes

Monday

  • Functions of Random Variables

  • Method of Transformation

  • Method of Moment Generating Functions

  • Method of Distribution Functions

Wednesday

  • Sampling Distributions

  • Central Limit Theorem

Reading

Day Reading
Monday’s Lecture MMS: 4.7
Wednesday’s Lecture MMS: 6.1

Homework

To be announced.

Important Concepts

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.

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 \(n\sigma^2\).

Resources

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Lecture Slides Videos
Monday Slides Video
Wednesday Slides Video