Week 4

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

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 $Y\le y$
3. Find $F_Y(y)=P(Y\le y)$ by integrating over the region of $Y\le y$
4. Find the density function $f_Y(y)=\frac{d{F}_Y(y)}{dy}$

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

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

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$T=\sum _{i=1}^n{X}_i$ 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