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: Recall the distribution of \(T\).
Since
\[ X_i\sim Exp(\beta), \] where \(\beta\) is the scale parameter, we have
\[ E(X_i)=\beta \] and
\[ Var(X_i)=\beta^2 \] Now,
\[ T=\sum_{i=1}^{n}X_i \] Therefore,
\[ T\sim Gamma(n,\beta) \] So,
\[ E(T)=n\beta \] and
\[ Var(T)=n\beta^2 \] Hence, option (A) is false.

Step 2: Check option (B).
If
\[ T\sim Gamma(n,\beta), \] then
\[ \frac{2T}{\beta}\sim \chi^2_{2n} \] Therefore, option (B) is true.

Step 3: Check option (C).
Let
\[ Y=\frac{\beta T}{2} \] Then, using scaling of expectation,
\[ E(Y)=\frac{\beta}{2}E(T) \] \[ E(Y)=\frac{\beta}{2}\cdot n\beta \] \[ E(Y)=\frac{n\beta^2}{2} \] Also,
\[ Var(Y)=\left(\frac{\beta}{2}\right)^2Var(T) \] \[ Var(Y)=\frac{\beta^2}{4}\cdot n\beta^2 \] \[ Var(Y)=\frac{n\beta^4}{4} \] Thus, option (C) is true.

Step 4: Check option (D).
For a gamma distribution with shape \(n>1\) and scale \(\beta\), the mode is
\[ (n-1)\beta \] Since
\[ T\sim Gamma(n,\beta), \] the mode of \(T\) is
\[ (n-1)\beta \] Therefore, option (D) is true.

Step 5: Final conclusion.
The true statements are
\[ \boxed{(B),\,(C)\text{ and }(D)} \]