# 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 |