A restaurant serves three fixed-price dinners costing 20, ...

A restaurant serves three fixed-price dinners costing $\$ 20, \$ 25,$ and $\$ 30 .$ For a randomly selected couple dining at this restaurant, let $X=$ the cost of the man's dinner and $Y=$ the cost of the woman's dinner. The joint pmf of $X$ and $Y$ is given in the following table: (a) Compute the marginal pmfs of $X$ and $Y .$ (b) What is the probability that the man's and the woman's dinner cost at most $\$ 25$ each? (c) Are $X$ and $Y$ independent? Justify your answer. (d) What is the expected total cost of the dinner for the two people? (e) Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find the message "You will receive as a refund the difference between the cost of the more expensive and the less expensive meal that you have chosen." How much does the restaurant expect to refund?

A restaurant serves three fixed-price dinners costing $\$ 20, \$ 25,$ and $\$ 30 .$ For a randomly selected couple dining at this restaurant, let $X=$ the cost of the man's dinner and $Y=$ the cost of the
woman's dinner. The joint pmf of $X$ and $Y$ is given in the following table:
(a) Compute the marginal pmfs of $X$ and $Y .$
(b) What is the probability that the man's and the woman's dinner cost at most $\$ 25$ each?
(c) Are $X$ and $Y$ independent? Justify your answer.
(d) What is the expected total cost of the dinner for the two people?
(e) Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find
the message "You will receive as a refund the difference between the cost of the more
expensive and the less expensive meal that you have chosen." How much does the
restaurant expect to refund?

Key Concepts

Transformation of Random Variables

This concept involves determining the distribution and expected value of a new random variable that is defined as a function of one or more existing random variables. It is useful when calculating quantities such as differences or sums derived from the original variables.

Expected Value

The expected value is a measure of the central tendency or average outcome of a random variable, computed as a weighted sum of all possible values with their associated probabilities. It is used to determine the long-run average result of repeated experiments or decisions.

Joint Probability Mass Function (joint pmf)

This concept describes how probabilities are assigned to the pairs of outcomes for two discrete random variables. It provides the complete probabilistic behavior of two variables taken together, indicating how likely each possible combination of outcomes is to occur.

Marginal Probability Mass Function

The marginal pmf is derived from the joint pmf by summing over the possible values of one variable to obtain the probabilities for the other variable. It shows the probability distribution of a single random variable without considering the other.

Independence of Random Variables

Two random variables are independent if the joint probability of any pair of outcomes is equal to the product of their respective marginal probabilities. This property implies that the outcome of one variable does not affect the outcome of the other.

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