What is the expected value of SN?

The expected value of a sum of random variables, denoted as E(Sn), is equal to the product of the number of variables, n, and the expected value of a single variable, µ. This fundamental property holds true regardless of the distribution of the individual variables.

In the context of statistics and probability theory, the expected value represents the long-run average value that one would expect to obtain from repeated trials of a random process. It is a measure of the central tendency of the distribution.

A Bernoulli random variable is a specific type of random variable that can take on only two possible values: 0 and 1. This binary nature makes it particularly useful in modeling situations where there are only two possible outcomes, such as success or failure, heads or tails, and so on.

The expected value of a Bernoulli random variable, denoted as E(Xn), is simply the probability of the variable taking on the value of 1, which is commonly referred to as the success probability. This means that if the success probability is µ, then E(Xn) = µ.

The property that the expected value of a Bernoulli random variable equals its success probability underscores the importance of the expected value as a measure of the average outcome. In situations where there are only two possible outcomes, the expected value provides a concise and informative summary of the distribution.