Question

If X and Y are Bernoulli random variables with parameters 0.2 and 0.35, which means X~Bo.2 and YBo.35. What is the Bernoulli parameter for the following random variables? If they are not Bernoulli, input -1 .x.y

Answer 1

1 - X

P(1 - X = 1) = P(X = 0) = 1 - 0.2 = 0.8

P(1 - X = 0) = P(X = 1) = 0.2

Thus, 1 - X ~ B_0.8

The parameter is 0.8

X²

P(X² = 1) = P(X = 1) = 0.2 as, X cannot be negative

P(X² = 0) = P(X = 0) = 0.8

X² ~ B_0.2

The parameter is 0.2

X.Y

P(X.Y = 1) = P(X = 1) * P(Y = 1) = 0.2 * 0.35 = 0.07

P(X.Y = 0) = 1- P(X.Y = 1) = 1 - 0.07 = 0.93

X.Y ~ B_0.07

The parameter is 0.07

|X - Y|

|X - Y| can take only values 0 and 1.

P(|X - Y| = 1) = P(X = 1) * P(Y = 0) + P(X = 0) * P(Y = 1) = 0.2 * (1 - 0.35) + (1 - 0.2) * 0.35 = 0.41

P(|X - Y| = 0) = 1 - 0.41 = 0.59

X + Y

X + Y = 2 when X = Y = 1. Thus, X + Y is not a Bernoulli random variable.

The answer is -1