Expected Value of a Discrete Random Variable

Given the probability distribution for the random variable x, representing the number of workouts in a week, find the expected value (mean) of x.

When tasked with finding the expected value of a random variable, it is important to recall that expected value is essentially the mean of the probability distribution. For a discrete random variable, the expected value is calculated as the sum of all possible values that the random variable can take, each multiplied by the probability of that value occurring. This operation essentially provides a weighted average of the possible outcomes, where the weights are the probabilities of each outcome.

In practical terms, the expected value gives a single summary measure of the random variable's long-term average if the underlying random experiment was repeated many times. It doesn't tell you about the variability or spread of the distribution, but rather offers a central point expected based on probabilities.

Understanding the concept of expected value is crucial as it lays the foundation for further statistical analysis. It is a key concept not only in probability but also in fields like finance, economics, and various engineering disciplines where decision-making under uncertainty is important. Mastering expected value in discrete settings will also aid in understanding continuous random variables, where integration rather than summation is used to derive expectations.