Question

The mini roulette is now played with an unfair wheel represented by a discrete random variable X with probability density function p(x) = P(X = x). Find the expected value of X.

Answer 1

The expected value of a discrete random variable X is calculated by multiplying each value by its probability and summing the products. This weighted average represents the long-term average of the variable when the experiment is repeated many times.

Step-by-step explanation:

The expected value, E(X), or mean μ, of a discrete random variable X can be calculated by multiplying each possible value of the random variable by its corresponding probability and then summing all those products together. The general formula for computing the expected value is given by E(X) = μ = Σ xP(x).

For instance, if we have a probability density function p(x) = P(X = x) that describes the likelihood of each outcome, we would construct a table listing all possible outcomes, x, alongside their corresponding probabilities, P(x). Then, we would add a new column to this table, x*P(x), representing the product of each outcome's value and its probability.

To calculate the expected value of X, we sum up all the values in the x*P(x) column. This sum is the long-term average or expected value of the random variable when the associated experiment is repeated a large number of times.