Chapter 7: Q. 7.10 (page 364)

Let $X$ be a Poisson random variable with mean $λ$ . Show that if $λ$ is not too small, then
$Var (\sqrt{X}) \approx . 25$
Hint: Use the result of Theoretical Exercise $7.4$ to approximate $E [\sqrt{X}]$ .

Short Answer

Expert verified

If $λ$ is not too small, then $Var (\sqrt{X}) \approx . 25$ is shown.

Step by step solution

Given Information

$λ$ is not too small.

Explanation

We know, $E [X] = 4, Var (X) = σ^{2}$

$E [g (x)] \approx f (μ) + \frac{g^{'} (μ)}{2} σ^{2}$

$E [\sqrt{X}] \approx \sqrt{λ} + \frac{λ}{2} \frac{d^{2}}{d λ^{2}}$

$= \sqrt{λ} + \frac{λ}{2} (\frac{- 1}{4 λ^{\frac{3}{2}}})$

Solved, $= \sqrt{λ} + \frac{λ}{2} (\frac{- 1}{8 λ^{\frac{1}{2}}})$

Explanation

If $E [X] = λ, Var (X) = λ$

Now, $Var (\sqrt{X}) = E [X] - (E [\sqrt{X}])^{2} = λ - (E [\sqrt{X}])^{2}$

$(E [\sqrt{X}])^{2} = λ + \frac{1}{64 λ} - \frac{1}{4}$

$Var (\sqrt{X}) = λ - λ - \frac{1}{64 λ} + \frac{1}{4} = \frac{1}{4} + \frac{1}{64 λ}$

Thus, $Var (\sqrt{X}) \approx \frac{1}{4} = 0.25$ .

Final answer

If $λ$ is not too small, then $Var (\sqrt{X}) \approx \frac{1}{4} = 0.25$ is shown.

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Let $X$ be a Poisson random variable with mean $λ$ . Show that if $λ$ is not too small, then
$Var (\sqrt{X}) \approx . 25$
Hint: Use the result of Theoretical Exercise $7.4$ to approximate $E [\sqrt{X}]$ .

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final answer

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Let Xbe a Poisson random variable with mean λ. Show that if λis not too small, thenVar(X)≈.25Hint: Use the result of Theoretical Exercise 7.4to approximateE[X].

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

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Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Let $X$ be a Poisson random variable with mean $λ$ . Show that if $λ$ is not too small, then
$Var (\sqrt{X}) \approx . 25$
Hint: Use the result of Theoretical Exercise $7.4$ to approximate $E [\sqrt{X}]$ .