Question

Let X equal the number of alpha particle emissions of carbon-14 that are counted by Geiger counter each second. Assume that the distribution of X is Poisson with mean 16. Let W equal the time in seconds before the seventh count is made
a) Give the distribution of W
b) Find \(P(W\leq 0.5)\)
Please, help

Answer 1

My attempt:
a) mean 16 means \(\lambda =16\), hence \(\theta =\dfrac{1}{\lambda}= \dfrac{1}{16}\)
so that the exponent distribution of W is f(x) = \(\huge f(x)=\dfrac{1}{\theta}e^{\frac{-x}{\theta}}\). In this case
\(\huge f(x) = 16 e^{\frac{-x}{16}}~~~0\leq x<\infty\)

Answer 2

b) first off, we need P(W>0.5) = \(\large^{\frac{-0.5}{\theta}}=e^{-8}= 0.000335\)
then \(P (W\leq 0.5) = 1-P(W >0.5) = 1-0.000335 =0.99965\)
But I am not sure about it.