# Random variables

AP®︎ Statistics,

A random variable is some outcome from a chance process, like how many heads will occur in a series of 20 flips (a discrete random variable), or how many seconds it took someone to read this sentence (a continuous random variable). We calculate probabilities of random variables, calculate expected value, and look what happens when we transform and combine random variables.

This course contains 7 segments:

Discrete random variables

Discrete random variables occur in situations where we can list every possible outcome of a process, and assign each outcome a probability in a probability model. For example, how many heads will we get in three flips? We'll learn how to make a probability distribution, find some probabilities, and calculate the mean (or expected value), variance, and standard deviation of a random variable.

Continuous random variables

Continuous random variables arise from situations where we measure some attribute on a scale — like height, weight, volume, temperature, or time. We'll learn how to identify a continuous random variable and calculate probabilities using area under density curves.

Transforming random variables

Transforming random variables produces predictable changes in the expected value and variance of the outcomes. Let's look at how changing units impacts the distribution of a discrete random variable.

Combining random variables

Combining random variables leads to some predictable and some less predictable distributions. Let's look at how adding and subtracting random variables can produce new distributions that help us solve some pretty interesting probability problems.

Binomial random variables

Binomial random variables are a special type of variable that comes up we repeat a process for a set number of independent trials, with each trial having the same probability of success, and we count how many successes we get at the end of the trials. For example, how many free-throws will a basketball player make in a series of 10 shots? Learn to identify these variables, use them to find probability, and calculate their mean and standard deviation.

Binomial mean and standard deviation formulas

Let's use what we've learned about combining random variables to create the mean and standard deviation formulas for a binomial random variable.

Geometric random variables

Geometric variables are similar but different from binomial variables. The key difference is that with a geometric variable, we count the number of trials it takes to get our first success. For example, how many attempts will it take me to make my first free-throw? Learn to calculate probabilities and the mean (expected value) for these variables.

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