Question

In a class of 60 students, each student belongs to exactly one of the three groups: A, B, or C. The number of students in each group is as follows:

Group A: 15 students
Group B: 20 students
Group C: 25 students
First, choose one of the 60 students at random, and let X be the size of that student's group (the number of students in the group). Next, choose one of the three groups at random, and let Y be the size of the selected group.

(a) Write down the probability mass function (pmf) for both X and Y.

(b) Calculate the expected value (mean) of X, denoted as E(X), and the expected value of Y, denoted as E(Y).

(c) Calculate the variance of X, denoted as Var(X), and the variance of Y, denoted as Var(Y).

Answer 1

The student needs to calculate the pmf for X and Y, the expected values E(X) and E(Y), and the variances Var(X) and Var(Y) of a random class selection scenario. This involves using probabilities based on the class sizes and calculating the means and variances accordingly.

Step-by-step explanation:

Probability Mass Function (pmf) and Expected Values

Let us tackle the problem part by part:

• Probability mass function for X: Since X is the size of the group to which a randomly chosen student belongs, we can define the pmf of X as: P(X = 15) = 15/60, P(X = 20) = 20/60, P(X = 25) = 25/60.
• Probability mass function for Y: Since Y is the size of a randomly chosen group, each group has an equal chance of being selected, so: P(Y = 15) = 1/3, P(Y = 20) = 1/3, P(Y = 25) = 1/3.
• Expected value for X (E(X)): E(X) is calculated as: E(X) = (15 * 15/60) + (20 * 20/60) + (25 * 25/60).
• Expected value for Y (E(Y)): E(Y) is calculated as: E(Y) = (15 * 1/3) + (20 * 1/3) + (25 * 1/3).
• Variance of X (Var(X)): Var(X) is the expectation of the squared deviation of X from its mean: Var(X) = ((15 - E(X))^2 * 15/60) + ((20 - E(X))^2 * 20/60) + ((25 - E(X))^2 * 25/60).
• Variance of Y (Var(Y)): Var(Y) is the expectation of the squared deviation of Y from its mean: Var(Y) = ((15 - E(Y))^2 * 1/3) + ((20 - E(Y))^2 * 1/3) + ((25 - E(Y))^2 * 1/3).

To solve the problems:

1. First, calculate the values of E(X) and E(Y) using the provided pmfs.
2. Then, use the calculated means to compute the variances Var(X) and Var(Y).