# 3: Probability Distributions

### Introduction

#### Learning objectives for this lesson

Upon completion of this lesson, you should be able to:

- distinguish between discrete and continuous random variables
- explain the difference between population, parameter, sample, and statistic
- determine if a given value represents a population parameter or sample statistic
- find probabilities associated with a discrete probability distribution
- compute the mean and variance of a discrete probability distribution
- find probabilities associated with a binomial distribution
- find probabilities associated with a normal probability distribution using the standard normal table
- determine the standard error for the sample proportion and sample mean
- apply the Central Limit Theorem properly to a set of continuous data

### Random Variables

A **random variable ** is numerical characteristic of each event in a sample space, or equivalently, each individual in a population.

Examples:

- The number of heads in four flips of a coin (a numerical property of each different sequence of flips).
- Heights of individuals in a large population.

Random variables are classified into two broad types

- A
**discrete random variable**has a countable set of distinct possible values. - A
**continuous random variable**is such that any value (to any number of decimal places) within some interval is a possible value.

**Examples of discrete random variable:**

- Number of heads in 4 flips of a coin (possible outcomes are 0, 1, 2, 3, 4).
- Number of classes missed last week (possible outcomes are 0, 1, 2, 3, ..., up to some maximum number)
- Amount won or lost when betting $1 on the Pennsylvania Daily number lottery

**Examples of continuous random variables:**

- Heights of individuals
- Time to finish a test
- Hours spent exercising last week.

**Note **: In practice, we don't measure accurately enough to truly see all possible values of a continuous random variable. For instance, in reality somebody may have exercised 4.2341567 hours last week but they probably would round off to 4. Nevertheless, hours of exercise last week is inherently a continuous random variable.