Question

Consider the sample space S = {a,b,c}, with equal probability for each outcome. Define the random variables X and Y by X(a) = -1, X(b) = 0, X(c) = 1, Y(a) = Y C) = 0, Y(6) = 1. Check that Var(X+Y) = Var(X) + Var(Y), but that X and Y are not independent.

Answer 1

To address the student's question, one must calculate the expected values and variances of the random variables X and Y, and verify that the variance of the sum equals the sum of the variances. It is also essential to check if the joint probabilities match the product of individual probabilities to determine the independence of X and Y.

Step-by-step explanation:

The student is asked to verify that Var(X+Y) = Var(X) + Var(Y), for random variables X and Y defined on a sample space S = {a,b,c}, and also to check if X and Y are independent. Since the equations involve variances and probabilities, this is a question of probability theory within the field of mathematics.

We begin by calculating the expected values (mean) and variances for X and Y. The expected value of a random variable X, denoted E(X), is the average or mean value that X takes on. Variance of X, denoted Var(X), measures how much the values of X vary from the mean, E(X).

The calculation of Var(X+Y) involves the sum of the variances of X and Y only if X and Y are independent. However, independence of two events A and B means that the occurrence of A does not affect the occurrence of B, which translates to P(A AND B) = P(A)P(B).

For variables X and Y to be independent, every pair of events X = x and Y = y must satisfy that condition. To check this, we calculate the joint probabilities and compare them to the products of the individual probabilities.