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: List the possible values of each variable.
Both $X$ and $Y$ are Bernoulli with values $0$ or $1$, each occurring with probability $\frac12$.
Step 2: Write down what $Z$ is.
Here $Z=XY$, a product. A product of zeros and ones is $1$ exactly when both factors are $1$, and $0$ otherwise.
Step 3: Find when $Z=1$.
We need $X=1$ and $Y=1$ simultaneously.
Step 4: Use independence to compute that probability.
$P(Z=1)=P(X=1)\,P(Y=1)=\frac12\times\frac12=\frac14$.
Step 5: Get the remaining probability.
$P(Z=0)=1-P(Z=1)=1-\frac14=\frac34$.
Step 6: State the full distribution.
So $Z$ takes value $1$ with probability $\frac14$ and value $0$ with probability $\frac34$.
\[ \boxed{P(Z=1)=\frac14,\;P(Z=0)=\frac34} \]