**Problem Statement:**2. Suppose alpha particle emissions of carbon-14 are counted by a Geiger counter each second. The emissions follow a Poisson process with a mean of 16 particles per second. Let \( W \) equal the time in seconds before the seventh particle is counted. a) Give the distribution of \( W \). b) Find \( P(W \leq 0.5) \).**Explanation:**- The problem refers to a scenario where alpha particles emitted from carbon-14 are detected by a Geiger counter. - The emissions are modeled as a Poisson process.- The Poisson process has a mean (or rate) of 16 particles per second.- \( W \) is the random variable representing the time until the seventh particle is observed.**Distribution of \( W \):**- Since \( W \) is defined as the waiting time for the seventh occurrence in a Poisson process, \( W \) follows a Gamma distribution.- Specifically, \( W \) has a Gamma distribution with shape parameter \( k = 7 \) and rate parameter \(\lambda = 16\).**Finding \( P(W \leq 0.5) \):**- This requires calculating the probability that the waiting time \( W \) is less than or equal to 0.5 seconds using the derived Gamma distribution parameters.

Given that alpha particle emissions of carbon-14 are counted by a Geiger counter each second. The…

Answered: 2. Suppose alpha particle emissions of…

MATLAB: An Introduction with Applications

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Publisher:Amos Gilat

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2. Suppose alpha particle emissions of carbon-14 are counted by a Geiger counter each second. The emissions follow a Poisson process with the mean of 16 particles per second. Let W equal the time in seconds before the seventh particle is counted. a) Give the distribution of W. b) Find P(W ≤ 0.5).