Chapter 7: Q. 7.51 (page 363)

Use Table $7.2$ to determine the distribution of $\sum_{i = 1}^{n} X_{i}$ when $X_{1}, \dots, X_{n}$ are independent and identically distributed exponential random variables, each having mean $1 / λ$ .

Short Answer

Expert verified

The identically distributed exponential random variables value are $\sum X_{i} ~ Gamma (n, λ)$ .

Step by step solution

Given Information

Determine the distribution of $\sum_{i = 1}^{n} X_{i}$ ,when $X_{1}, \dots, X_{n}$ are independent.

Explanation

We have that $X_{1}, \dots, X_{n}$ have distribution $Expo (λ)$ and that they are independent.

So, they have all common MGF

$M (t) = \frac{λ}{λ - t}$

Define $Y = \sum_{i = 1}^{n} X_{i}$ we have that,

$M_{Y} (t) = E (e^{t Y}) = E (e^{t \sum_{i = 1}^{n} X_{i}})$ $= \prod_{i = 1}^{n} E (e^{t X_{i}}) = M (t)^{n}$ .

Explanation

Where the third and fourth equality hold since variables $X_{i}$ are independent and equally distributed.

So, we have that,

$M_{Y} (t) = {(\frac{λ}{λ - t})}^{n}$

Now, using the calculated MGF of $Y$ , we can recognize that $Y$ has Gamma distribution with parameters $n$ and $λ$ .

Final answer

We can recognize that $Y$ has Gamma distribution with parameters $n$ and $λ$ value are $\sum X_{i} ~ Gamma (n, λ)$ .

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Use Table $7.2$ to determine the distribution of $\sum_{i = 1}^{n} X_{i}$ when $X_{1}, \dots, X_{n}$ are independent and identically distributed exponential random variables, each having mean $1 / λ$ .

Short Answer

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Use Table 7.2 to determine the distribution of∑i=1nXi when X1,…,Xnare independent and identically distributed exponential random variables, each having mean1/λ.

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final answer

One App. One Place for Learning.

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Use Table $7.2$ to determine the distribution of $\sum_{i = 1}^{n} X_{i}$ when $X_{1}, \dots, X_{n}$ are independent and identically distributed exponential random variables, each having mean $1 / λ$ .