Question

Let \(X\) and \(Y\) be two independent identically distributed Bernoulli random variables with \[ P(X=1)=\frac12, \qquad P(X=0)=\frac12. \] If \(Z=XY\), then distribution of \(Z\) is:

• A. \(P(Z=1)=\frac23,\;P(Z=0)=\frac13\)
• B. \(P(Z=1)=\frac12,\;P(Z=0)=\frac12\)
• C. \(P(Z=1)=\frac14,\;P(Z=0)=\frac34\)
• D. \(P(Z=1)=\frac13,\;P(Z=0)=\frac23\)

Hint:

For Bernoulli variables, the product equals 1 only when both variables are equal to 1.

Answer 1

The Correct Option is C

Step 1: Determine when \(Z=1\).
Since \[ Z=XY, \] \(Z=1\) only when \[ X=1 \quad\text{and}\quad Y=1. \] Because \(X\) and \(Y\) are independent, \[ P(Z=1) = P(X=1)P(Y=1) = \frac12\times\frac12 = \frac14. \]

Step 2: Find \(P(Z=0)\).
\[ P(Z=0) = 1-\frac14 = \frac34. \] Hence \[ \boxed{P(Z=1)=\frac14,\quad P(Z=0)=\frac34}. \]