Question

A radioactive mass emits particles according to a Poisson process at a mean rate of 2 per second. Let T be the waiting time, in seconds, between emissions. a. What is the mean waiting time? b. What is the median waiting time? c. Find P(T > 2). d. Find P(T < 0.1).

Answer 1

Answer with Step-by-step explanation:

We are given that

`\lambda=2`

a.Mean waiting time,
`E(T)=(1)/(\lambda)`

`E(T)=(1)/(2)=0.5 s`

b.Median waiting time,
`P(T<t_(md))=E(T)=0.5`

`1-e^{-\lambda t_(md)}=0.5`

Where
`P(T<t)=1-e^(-\lambda t)`

`e^{-\lambda t_(md)}=1-0.5=0.5`

`e^{-2t_(md)}=0.5`

`-2t_(md)=ln(0.5)`

`-2t_(md)=-0.693`

`t_(md)=(0.693)/(2)=0.3465 s`

c.P(T>2)=
`1-P(T< 2)=1-(1-e^(-2* 2))`

`P(T>2)=e^(-4)=0.018`

d.
`P(T<0.1)=1-e^(-2* 0.1)=1-e^(-0.2)=0.181`