A random variable is a mathematical concept used in probability theory to represent the possible outcomes of a random process or experiment. It is typically denoted by a capital letter, such as X, and can take on numerical values.
The expected value of a random variable is a measure of the central tendency of its distribution. It represents the average value of the random variable if the experiment were to be repeated an infinite number of times.
The expected value of a random variable is calculated by multiplying each possible value of the random variable by its corresponding probability and then summing all of these products together. This can be represented mathematically as E[X] = Σx P(X=x).
Some examples of random variables and their expected values include the number of heads obtained when flipping a fair coin, the roll of a die, and the height of a randomly selected person from a population. In these examples, the expected value would be 0.5, 3.5, and the average height of the population, respectively.
Functions of random variables are also random variables themselves, and their expected values can be calculated using the same formula. Additionally, the expected value of a linear function of a random variable is equal to the function of the expected value of the random variable, which is known as the linearity property.