# Week 5

This week, we will discuss continuous random variables and their properties.
Published

February 20, 2023

## Learning Outcomes

### First Lecture

• Continuous Random Variables

### Second Lecture

• Expected Values
• MGF

## Important Concepts

### First Lecture

#### Continuous Variables

A random variable $$X$$ is considered continuous if the $$P(X=x)$$ does not exist.

##### CDF

The cumulative distribution function of $$X$$ provides the $$P(X\leq x)$$, denoted by $$F(x)$$, for the domain of $$X$$.

Properties of the CDF of $$X$$:

1. $$F(-\infty)\equiv \lim_{y\rightarrow -\infty}F(y)=0$$
2. $$F(\infty)\equiv \lim_{y\rightarrow \infty}F(y)=1$$
3. $$F(x)$$ is a nondecreaseing function
##### PDF

The probability density function of the random variable $$X$$ is given by

$f(x)=\frac{dF(x)}{d(x)}=F^\prime(x)$

wherever the derivative exists.

Properties of pdfs:

1. $$f(x)\geq 0$$
2. $$\int^\infty_{-\infty}f(x)dx=1$$
3. $$P(a\leq X\leq b) = P(a<X<b)=\int^b_af(x)dx$$
##### Commonly Used Distributions
Distribution Parameters PDF
Uniform $$a$$ and $$b$$ $$\frac{1}{b-a}$$
Normal $$\mu$$ and $$\sigma^2$$ $$\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}$$
Exponential $$\lambda$$ $$\lambda e^{-\lambda x}$$
Gamma $$\alpha$$ and $$\beta$$ $$\frac{x^{\alpha-1}e^{-x/\beta}}{\beta^\alpha\Gamma(\alpha)}$$
Beta $$\alpha$$ and $$\beta$$ $$\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int^1_0x^{\alpha-1}(1-x)^{\beta-1}dx}$$

### Expected Value

The expected value is the value we expect when we randomly sample from population that follows a specific distribution. The expected value of a discrete random variable is $$Y$$ with PDF of $$f(y)$$ is

$E(Y)=\int_y yf(y)dy$

### Variance

The variance represents the variation of a random variable. The variance for a random variable Y is

$Var(Y) = E[\{Y-E(Y)\}^2]$

## Resources

### First Lecture

Slides Videos
Slides Video 001 Video 002

### Second Lecture

Slides Videos
Slides Video 001 Video 002