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: Rates.
\(U_1\) rate 1, \(U_2\) rate 2, \(U_3\) rate 3.

Step 2: X=Y condition.
Apart from probability-zero ties, \(X=Y\) iff both \(U_1>U_3\) and \(U_2>U_3\).

Step 3: Integral.
\[ P(X=Y)=\int_0^\infty f_3(t)P(U_1>t)P(U_2>t)dt=\int_0^\infty 3e^{-3t}e^{-t}e^{-2t}dt=\int_0^\infty 3e^{-6t}dt \]

Step 4: Evaluate.
\[ 3\times\frac{1}{6}=\frac{1}{2} \]

Final Answer: \[ \boxed{P(X=Y)=0.50} \]