Question

Let \(X_1,X_2,\ldots,X_n\) be a random sample of size \(n\;(n>1)\) from an \(Exp(\beta)\) distribution, where \(\beta>0\). If \(T=\sum_{i=1}^{n}X_i\), then which of the following statements is/are true?

• A. \(T\) follows a gamma distribution with mean \((n-1)\beta\) and variance \((n-1)\beta^2\)
• B. \(\dfrac{2T}{\beta}\) follows a \(\chi^2_{2n}\) distribution
• C. \(\dfrac{\beta T}{2}\) follows a gamma distribution with mean \(\dfrac{n\beta^2}{2}\) and variance \(\dfrac{n\beta^4}{4}\)
• D. The mode of \(T\) is \((n-1)\beta\)

Hint:

The sum of \(n\) independent exponential random variables with scale parameter \(\beta\) follows a gamma distribution with shape \(n\) and scale \(\beta\).

Answer 1

The Correct Option is B, C, D

Step 1: Distribution of $T$.
$T=\sum X_i$ with $X_i\sim Exp(\beta)$ is $Gamma(n,\beta)$, so $E(T)=n\beta$ and $\mathrm{Var}(T)=n\beta^2$.

Step 2: Check (A).
(A) claims mean $(n-1)\beta$ and variance $(n-1)\beta^2$, but the true values are $n\beta$ and $n\beta^2$. (A) fails.

Step 3: Check (B).
For $Gamma(n,\beta)$, $\frac{2T}\beta\sim\chi^2_{2n}$. (B) holds.

Step 4: Check (C) and (D).
For $Y=\frac{\beta T}2$, $E(Y)=\frac\beta2\cdot n\beta=\frac{n\beta^2}2$ and $\mathrm{Var}(Y)=\frac{\beta^2}4\cdot n\beta^2=\frac{n\beta^4}4$, so (C) holds. The mode of $Gamma(n,\beta)$ with $n>1$ is $(n-1)\beta$, so (D) holds.

Step 5: Collect.
\[ \boxed{(B),(C),(D)} \]