Chapter 4: Q.4.19 (page 171)

Show that $X$ is a Poisson random variable with parameter $λ$ , then
$E [X^{n}] = λ E [(X + 1)^{n - 1}]$
Now use this result to compute $E [X^{3}]$ .

Short Answer

Expert verified

$E (X^{3}) = λ^{3} + 3 λ^{2} + λ$

Step by step solution

Step 1: Given information

Given in the question that $X$ is a Poisson random variable with parameter $λ$ , then $E [X n] = λ E [(X + 1) n - 1]$ . We need to find $E [X 3]$

Step 2: Explanation

Using the theorem about the mean of function of random variable,

we have that

$E (X^{n}) = \sum_{k = 0}^{\infty} k^{n} \frac{λ^{k}}{k!} e^{- λ} = \sum_{k = 1}^{\infty} k^{n} \frac{λ^{k}}{k!} e^{- λ} = λ \sum_{k = 1}^{\infty} k^{n} \frac{λ^{k - 1}}{k!} e^{- λ}$

$= λ \sum_{k = 1}^{\infty} k^{n - 1} \frac{λ^{k - 1}}{(k - 1)!} e^{- λ} = λ \sum_{k = 0}^{\infty} (k + 1)^{n - 1} \frac{λ^{k}}{k!} e^{- λ} = λ E ((X + 1)^{n - 1})$

which had to be proved. Using that, we have that

$E (X^{3}) = λ E ((X + 1)^{2}) = λ E (X^{2} + 2 X + 1) = λ (E (X^{2}) + 2 E (X) + 1)$

Now, use that $E (X^{2}) = Var (X) + E X^{2} = λ + λ^{2}$

and that $E X = λ$ ,

so we have that the expression above is equal to

$= λ (λ + λ^{2} + 2 λ + 1) = λ^{3} + 3 λ^{2} + λ$

Step 3:Final answer

$E (X^{3}) = λ^{3} + 3 λ^{2} + λ$

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Show that $X$ is a Poisson random variable with parameter $λ$ , then
$E [X^{n}] = λ E [(X + 1)^{n - 1}]$
Now use this result to compute $E [X^{3}]$ .

Short Answer

Step by step solution

Step 1: Given information

Step 2: Explanation

Step 3:Final answer

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Show that Xis a Poisson random variable with parameter λ, thenEXn=λE(X+1)n-1Now use this result to compute EX3.

Short Answer

Step by step solution

Step 1: Given information

Step 2: Explanation

Step 3:Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Show that $X$ is a Poisson random variable with parameter $λ$ , then
$E [X^{n}] = λ E [(X + 1)^{n - 1}]$
Now use this result to compute $E [X^{3}]$ .