Question

Let $X_1, X_2, \ldots, X_n$ be a random sample from $U(1,2)$ and $Y_1, Y_2, \ldots, Y_n$ be a random sample from $U(0,1)$. Suppose the two samples are independent. Define \[ Z_i = \begin{cases} 1, & \text{if } X_i Y_i < 1, \\ 0, & \text{otherwise}, \end{cases} i = 1,2, \ldots, n. \] If $\lim_{n \to \infty} P\left(|\frac{1}{n} \sum_{i=1}^n Z_i - \theta| < \epsilon\right) = 1$ for all $\epsilon > 0$, then $\theta$ equals 
 

• A. $\dfrac{1}{4}$
• B. $\dfrac{1}{2}$
• C. $\log_e \dfrac{3}{2}$
• D. $\log_e 2$

Hint:

For uniform random pairs $(X,Y)$, the probability $P(XY < c)$ is found using geometric integration over the valid rectangular support region.

Answer 1

The Correct Option is D

Step 1: Definition of $\theta$. 
Since the law of large numbers applies, $\theta = E(Z_i) = P(X_i Y_i < 1)$.

Step 2: Compute the probability. 
The joint pdf is uniform over $x \in [1,2]$, $y \in [0,1]$. We need the area under the region $xy < 1$. For $x \in [1,2]$, $y < \frac{1}{x}$.

Step 3: Set up the integral. 
\[ \theta = \int_{x=1}^{2} \int_{y=0}^{1/x} dy \, dx = \int_{1}^{2} \frac{1}{x} \, dx = \log_e 2 - \log_e 1 = \log_e 2. \] 

Step 4: Conclusion. 
\[ \boxed{\theta = \log_e 2} \]