Poisson Distribution - why are these different?

[spitz]

Homework Statement

$X(t)$ is a Poisson process with $\lambda=0.2$ events per second. What is the probability of zero events in $45$ seconds?

2. The attempt at a solution

$\frac{45}{0.2}=225$ ($0.2$ second intervals)

so $P[X=0]$ in $225$ consecutive intervals is:

$\left(e^{-0.2}\right)^{225} = 2.86 \times 10^{-20}$

or

$\lambda = (45)(0.2)=9$ events per $45$ seconds gives:

$e^{-9} = 1.23 \times 10^{-4}$

Both are approx. zero, but they are way different. How come?

 

[Ray Vickson]

Homework Statement

$X(t)$ is a Poisson process with $\lambda=0.2$ events per second. What is the probability of zero events in $45$ seconds?

2. The attempt at a solution

$\frac{45}{0.2}=225$ ($0.2$ second intervals)

so $P[X=0]$ in $225$ consecutive intervals is:

$\left(e^{-0.2}\right)^{225} = 2.86 \times 10^{-20}$

or

$\lambda = (45)(0.2)=9$ events per $45$ seconds gives:

$e^{-9} = 1.23 \times 10^{-4}$

Both are approx. zero, but they are way different. How come?

The 0.2 is---as you stated originally--- the number of events per second; there is no 0.2 sec anywhere in the problem! In other words, 0.2/sec = (1/5)/sec = 1 per 5 sec. This is a good illustration of the old advice: always check your units (sec ≠ per sec).

RGV