Let X equal the number of alpha particle emissions of

Probability and Statistical Inference | 9th Edition | ISBN: 9780321923271 | Authors: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

ISBN: 9780321923271 41

Solution for problem 12E Chapter 3.2

Probability and Statistical Inference | 9th Edition

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Problem 12E

Let X equal the number of alpha particle emissions of carbon-14 that are counted by a Geiger counter each second. Assume that the distribution of X is Poisson with mean 16. LetW equal the time in seconds before the seventh count is made.

(a) Give the distribution of W.

(b) Find P(W ≤ 0.5). Hint: Use Equation 3.2-1 with λw = 8.

Step-by-Step Solution:

Step 1 of 3:

Given that X denotes the number of carbon particles emission of carbon-14 in each second and W denotes the time in seconds before the seventh count is made.

Also it is given that X has Poisson distribution with parameter 16. That is X~P(16).

We have to find the distribution of W and then using the given equation we have to find the required probability.

Step 2 of 3:

(a)

Here we have to identify the distribution of W.

W has a gamma distribution with parameter $\alpha{}$ =7 and $\theta{}$ = $\frac{1}{\lambda{}}$ ,that is $\theta{}=$ $\frac{1}{16}$

Symbolically we can write it as W~G( $\alpha{}$ =7, $\theta{}=$ $\frac{1}{16}$ ).

Thus,

f(W)= $\frac{{w}^{\alpha{}-1}{e}^{-\frac{x}{\theta{}}}}{\Gamma{}(\alpha{}){\theta{}}^{\alpha{}}}$

= $\frac{{w}^{6}{e}^{-16w}}{6!*{(\frac{1}{16})}^{6}}$

= $\frac{{w}^{6}{e}^{-16w}{(16)}^{7}}{6!}$

Step 3 of 3

Chapter 3.2, Problem 12E is Solved

Textbook: Probability and Statistical Inference

Edition: 9

Author: Robert V. Hogg, Elliot Tanis, Dale Zimmerman

ISBN: 9780321923271

This textbook survival guide was created for the textbook: Probability and Statistical Inference , edition: 9. The answer to “Let X equal the number of alpha particle emissions of carbon-14 that are counted by a Geiger counter each second. Assume that the distribution of X is Poisson with mean 16. LetW equal the time in seconds before the seventh count is made.(a) Give the distribution of W.(b) Find P(W ? 0.5). Hint: Use Equation 3.2-1 with ?w = 8.” is broken down into a number of easy to follow steps, and 60 words. This full solution covers the following key subjects: distribution, let, geiger, carbon, count. This expansive textbook survival guide covers 59 chapters, and 1476 solutions. The full step-by-step solution to problem: 12E from chapter: 3.2 was answered by , our top Statistics solution expert on 07/05/17, 04:50AM. Since the solution to 12E from 3.2 chapter was answered, more than 722 students have viewed the full step-by-step answer. Probability and Statistical Inference was written by and is associated to the ISBN: 9780321923271.