# Week 3

## Learning Outcomes

### First Lecture

- Describe Discrete Distributions

### Second Lecture

Expected Values

Variance

Properties of Expected Values

## Important Concepts

### First Lecture

#### Discrete Variables

A random variable is considered to be discrete if it can only map to a finite or countably infinite number of distinct values.

##### PMF

The probability mass function of discrete variable can be represented by a formula, table, or a graph. The Probability of a random variable Y can be expressed as \(P(Y=y)\) for all values of \(y\).

##### CDF

The cumulative distribution function provides the \(P(Y\leq y)\) for a random variable \(Y\).

##### Commonly Used Distributions

Distribution | Parameter(s) | PMF \(P(Y=y)\) |
---|---|---|

Bernoulli | \(p\) | \(p\) |

Binomial | \(n\) and \(p\) | \((^n_y)p^y(1-p)^{n-y}\) |

Geometric | \(p\) | \((1-p)^{y-1}p\) |

Negative Binomial | \(r\) and \(p\) | \((^{y-1}_{r-1})p^{r-1}(1-p)^{y-r}\) |

Hypergeometric | \(N\), \(n\), and \(r\) | \(\frac{(^r_y)(^{N-r}_{n-y})}{(^N_n)}\) |

Poisson | \(\lambda\) | \(\frac{\lambda^y}{y!} e^{-\lambda}\) |

### 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 PMF of \(P(y)\) is

\[ E(Y)=\sum_y yP(y) \]

#### Expected Value Properties

Special properties of the expected value:

\(E(c)=c\), where \(c\) is constant

\(E\{cg(Y)\}=cE\{g(Y)\}\)

\(E\{g_1(Y)+\cdots+g_n(Y)\}=E\{g_1(Y)\}+\cdots+E\{g_n(Y)\}\)

#### 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 |