Question

Let \(U_1,U_2,U_3\) be three independent exponential random variables such that \(U_k\) has the following probability density function
\[ f_k(x)=\begin{cases}ke^{-kx}&\text{if }x>0\\0&\text{otherwise,}\end{cases}\quad k=1,2,3. \]
Let
\[ X=\min\{U_1,U_3\}\quad\text{and}\quad Y=\min\{U_2,U_3\}. \]
Then \(P(X=Y)\) equals ______ (rounded off to two decimal places).

Hint:

\(X=Y\) happens (ignoring probability zero ties) exactly when \(U_3\) is smaller than both \(U_1\) and \(U_2\); integrate over the density of \(U_3\).

Answer 1

Correct Answer: 0.5

Step 1: Racing clocks.
$X=Y$ happens iff $U_3$ is the smallest of the three.

Step 2: Competing exponentials fact.
\[ P(U_i=\min\{U_1,U_2,U_3\})=\frac{\lambda_i}{\lambda_1+\lambda_2+\lambda_3} \]

Step 3: Apply for U3.
\[ P(U_3=\min)=\frac{3}{1+2+3}=\frac{1}{2} \]

Step 4: Conclude.
\[ P(X=Y)=0.50 \]
\[ \boxed{P(X=Y)=0.50} \]