Question

Ex 3

I he interval" s Ol he can butcomě in an be time, distance, area, volume, or some similar unit. interval. Definition: A random variable X is said to have a Poisson distribution and is referred to as a Poisson random variable, if and only if its probability distribution is given by for x 0, 1,2,... where 1 the average number of outcomes occurred per unit "interval" .If X Poisson (2), then E(X) Var(X)a Example 3: Find (i) Two customers arrive in any particular minute. (ii) Two customers arrive in any 2 minutes. (ii) Two customers arrive in any 30-second period.

Answer 1

The average rate, stated as Lambda in the problem, is given as 3.4 per minute. A beautiful aspect of the Poisson formula is that it can be applied with any time interval, apart from that mentioned. For ex, here the first part of problem asks us to compute in the same time interval of a minute. But the other two parts have different time intervals. We can do the linear translation of average in order to compute the probabilities, as follows

$\textup{(i) }\lambda =3.4.\textup{ Hence, }\mathbb{P}(X=2)=\frac{3.4^{2}\times e^{-3.4}}{2!}\approx 0.193$

$\textup{(ii) For 2 minutes, }\lambda =6.8.\textup{ Hence, }\mathbb{P}(X=2)=\frac{6.8^{2}\times e^{-6.8}}{2!}\approx 0.02575$

$\textup{(iii) For 0.5 minutes, }\lambda =1.7.\textup{ Hence, }\mathbb{P}(X=2)=\frac{1.7^{2}\times e^{-1.7}}{2!}\approx 0.264$

$\textbf{Principle used above: }\textup{If the average rate is }\lambda \textup{ for time interval T in Poisson distribution,}$

$\textup{then the Poisson formula still holds true for time interval kT with rate k}\lambda$