# 4.7 Introduction to Random Variables and Probability Distributions

Kanya Shah

Focus on the big picture in terms of what context the variables exist in.
A random variable takes numerical values that describe the outcomes of a chance process. The probability distribution of a random variable gives its possible values and their probabilities. To assign labels to random variables, we use capital letters (X or Y).

## Two types of random variables

A continuous random variable can take any value in an interval on the number line so this includes whole numbers and decimals for value of X. Generally, you use a density curve to find the probability of a continuous variable and the probability usually applies to an interval rather than individual values.
A discrete random variable X takes a fixed set of possible values with gaps between them (cannot include decimals so, whole numbers only). *For a probability distribution to be valid, each probability must be between 0 and 1, inclusive. Also, the sum of the probabilities must add to 1.

## Critical Concept

When calculating probability for discrete random variables, always think about whether you should include the boundary value in your calculations. Make sure you understand how to calculate the probability of a discrete random variable given P(Xn) and P(X=n). The wording is usually confusing so draw yourself a mini probability distribution chart to figure out whether you should/shouldn’t include the boundary value. This will help you when there are phrases like at least, no more than, greater than, etc.

### Example of a Probability Distribution for a discrete random variable

 Value x1 x2 x3 x4 Probability p1 p2 p3 p4
You need to know how to represent a discrete random variable as a histogram or in a table. For the histogram, use the discrete random variable as the x axis values and the probabilities for the y axis.

### Analyzing the Shape of a Discrete Random Variable Graph

When describing the shape of a discrete random variable, talk about whether the graph is roughly symmetric, double/single peaked, and right/left skewed. Don’t forget to mention the center (mean) and measure of variability (standard deviation). These interpretations will allow you to make conclusions.

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