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Get Full Access to Probability And Statistical Inference - 9 Edition - Chapter 2.5 - Problem 5e
Get Full Access to Probability And Statistical Inference - 9 Edition - Chapter 2.5 - Problem 5e

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# Use the result of Exercise 2.5-5 to find the mean and

ISBN: 9780321923271 41

## Solution for problem 5E Chapter 2.5

Probability and Statistical Inference | 9th Edition

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Problem 5E

Let the moment-generating function M(t) of X exist for $$-h<t<h$$. Consider the function $$R(t)=\ln M(t)$$. The first two derivatives of R(t) are, respectively,

$$R^{\prime}(t)=\frac{M^{\prime}(t)}{M(t)} \text { and } R^{\prime \prime}(t)=\frac{M(t) M^{\prime \prime}(t)-\left[M^{\prime}(t)\right]^{2}}{[M(t)]^{2}}$$

Setting , show that

(a) .

(b) .

Equation Transcription:

.

Text Transcription:

-h < t < h

R(t) = ln M(t)

R prime prime (t) = M(t)M prime prime (t) - [M prime (t)]^2/[M(t)]^2

Nu = R prime (0)

Sigma^2 = R prime prime (0)

Step-by-Step Solution:

Step 1 of 6:

From exercise 2.5 - 5.

and

Put t = 0.

Then,

=

=

= .

Therefore,  =

=

=

=

Therefore,

=  and

is the moment generating function.

Step 2 of 5

Step 3 of 5

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