Question

Let \(X_1,X_2,\ldots,X_{10}\) be a random sample of size \(10\) from an \(Exp(\beta)\) distribution, where \(\beta>0\) is an unknown parameter. Let

• A. \(P\) is correct and \(Q\) is NOT correct
• B. \(P\) is NOT correct and \(Q\) is correct
• C. Both \(P\) and \(Q\) are correct
• D. Neither \(P\) nor \(Q\) is correct

Hint:

For \(X\sim Exp(\beta)\) with scale parameter \(\beta\), the MLE is \(\hat{\beta}=\bar{X}\), and by invariance property, the MLE of any function \(g(\beta)\) is \(g(\hat{\beta})\).

Answer 1

The Correct Option is A

Step 1: Identify the MLE of $\beta$.
For $Exp(\beta)$ with scale $\beta$, the MLE is the sample mean, here $T$.

Step 2: Check $P$ via the median.
Solving $1-e^{-m/\beta}=\tfrac12$ gives the median $m=\beta\ln2$. By invariance the MLE of the median is $T\ln2$, so $P$ is correct.

Step 3: Check $Q$.
We want the MLE of $P(X_1<2)=1-e^{-2/\beta}$, which becomes $1-e^{-2/T}$.

Step 4: Spot the mismatch.
Statement $Q$ gives $e^{-2/T}$, which is the MLE of $P(X_1>2)$, not $P(X_1<2)$. So $Q$ is not correct.

Step 5: Conclude.
$P$ right, $Q$ wrong, option (A).
\[ \boxed{(A)} \]