Week 5
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 \(X\):
- \(F(-\infty)\equiv \lim_{y\rightarrow -\infty}F(y)=0\)
- \(F(\infty)\equiv \lim_{y\rightarrow \infty}F(y)=1\)
- \(F(x)\) 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:
- \(f(x)\geq 0\)
- \(\int^\infty_{-\infty}f(x)dx=1\)
- \(P(a\leq X\leq b) = P(a<X<b)=\int^b_af(x)dx\)
Commonly Used Distributions
| Distribution | Parameters | |
|---|---|---|
| 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
\[ E(Y)=\int_y yf(y)dy \]
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 |