Random variables can be any outcomes from some chance process, like how many heads will occur in a series of 20 flips. We calculate probabilities of random variables and calculate expected value for different types of random variables.
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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?

Transforming and combining random variables produces predictable changes in the expected value and variance of the outcomes.

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 shots will a basketball player make in a series of 10 shots?

The Poisson distribution is related to the binomial setting, but looks at the number of successes over time instead of over many repeated trials.