Question

Suppose x and y are discrete iid random variables, each taking values 0 and 1 with equal probability. Let Z=x+y. What is the probability mass function of Z?

(a) P(Z=0)=1/4​, P(Z=1)=1/2​, P(Z=2)=1/4​
(b) P(Z=0)=1/2​, P(Z=1)=1/2​, P(Z=2)=0
(c) P(Z=0)=0, P(Z=1)=1​/2, P(Z=2)=1​/2
(d) P(Z=0)=1/4​, P(Z=1)=1/2​, P(Z=2)=0

Answer 1

The probability mass function of Z can be found using the basic Addition Rule and the probabilities of X and Y taking their respective values. The probability mass function of Z is P(Z=0) = 1/4, P(Z=1) = 1/2, P(Z=2) = 1/4.

Step-by-step explanation:

To find the probability mass function of Z, we need to consider all possible values of Z and calculate their respective probabilities. Since X and Y are discrete and have equal probabilities of taking values 0 or 1, we can calculate the probability of Z using the basic Addition Rule:

P(Z) = P(X=0 AND Y=0) + P(X=0 AND Y=1) + P(X=1 AND Y=0) + P(X=1 AND Y=1)

Since X and Y are independent, each term in the above equation is the product of the probabilities of X and Y taking their respective values:

P(Z=0) = P(X=0) * P(Y=0) = 1/2 * 1/2 = 1/4

P(Z=1) = P(X=0) * P(Y=1) + P(X=1) * P(Y=0) = 1/2 * 1/2 + 1/2 * 1/2 = 1/2

P(Z=2) = P(X=1) * P(Y=1) = 1/2 * 1/2 = 1/4

Therefore, the probability mass function of Z is P(Z=0) = 1/4​, P(Z=1) = 1/2​, P(Z=2) = 1/4​