Week 5

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

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.


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

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 \(\beta\) \(\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int^1_0x^{\alpha-1}(1-x)^{\beta-1}dx}\)

Second Lecture

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

\[ E(Y)=\int_y yf(y)dy \]


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

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


First Lecture

Slides Videos
Slides Video 001 Video 002

Second Lecture

Slides Videos
Slides Video 001 Video 002