Final answer:
The expected value of X, which represents the number of doughnuts purchased by a customer, can be calculated by multiplying each possible value of X by its corresponding probability and summing them up. In this case, since the price of each doughnut is $1.50, the expected value of X is $1.50. The correct answer is B. $1.50.
Step-by-step explanation:
The expected value of a random variable can be calculated by multiplying each possible value of the variable by its corresponding probability and summing them up. In this case, the random variable X represents the number of doughnuts purchased by a customer, and the price of each doughnut is $1.50. Since the price remains the same for all possible values of X, the expected value can be calculated as follows:
E(X) = 1.50 * P(X = 0) + 1.50 * P(X = 1) + 1.50 * P(X = 2) + ...
Since the sum of probabilities for all possible values of X should be equal to 1, we can write the equation as:
E(X) = 1.50 * (P(X = 0) + P(X = 1) + P(X = 2) + ...)
Since each individual's purchase of doughnuts is independent and can take any non-negative integer value, the distribution for X is a discrete probability distribution known as the Poisson distribution.
The expected value of a Poisson random variable is equal to its rate parameter, which in this case is the average number of doughnuts purchased. Therefore, the expected value of X is $1.50, which corresponds to option B.