# Thread: Poissons & Poissons Random Variable

1. ## Poissons & Poissons Random Variable

Hi guys thanks for your help earlier.

Have just a few more that perhaps you could help me check.

First of all i have a question that states :

Suppose that the number of cracks per concrete specimen for a particular type of
cement mix, X say, has approximately a Poisson probability distribution. Further-
more, assume that the average number of cracks per specimen is 2.5.

q1) what are the chances of the number of cracks being 5 given that the number of cracks is an even number.

A) now obviously this is impossible so there is 0 chance.

the next bit is where it gets tricky.

Q2)
Find the probability that a randomly selected concrete specimen has 6 cracks
given that the number of cracks on the specimen is an even number.

A) now if you simply put the numbers into poissons you come out with 0.02783

now i was told by my lecturer that this is not correct as i have not taken the chances of it being even into account. So any help on this would be helpful.

also i have a question and have no idea where to start so any specific points or help would be really greatful.

So the real toughy is :

In a photoelectric detector, the number of photoelectrons
Y produced in time t

depends on the (normalized) incident energy
X. If X were constant, say, X = x,

Y
would be a Poisson random variable with mean x, but as real light sources {
except for gain-stabilized lasers { do not emit constant energy signals,
X must be
treated as a random variable. In certain situations
X takes ¯nite number of values
with probability mass function given by

P
(X = xi) = pi; i = 1; : : : ; m:

Show that the average value of photoelectrons
Y equals to the average value of the
incident energy
X, i.e.

E(Y ) = E(X):

now im completely clueless on how to do this so some help would really be helpful as im lost

Cheers guys and gals!

2. Originally Posted by mxmadman_44
the next bit is where it gets tricky.

Q2) [FONT=CMR12]
Find the probability that a randomly selected concrete specimen has 6 cracks
given that the number of cracks on the specimen is an even number.

A) now if you simply put the numbers into poissons you come out with 0.02783

now i was told by my lecturer that this is not correct as i have not taken the chances of it being even into account. So any help on this would be helpful.
Use Bayes' theorem:

$P(n=6|n \mbox{ even})=\dfrac{P(n \mbox{ even}|n=6)P(n=6)}{p(n \mbox{ even})}=\dfrac{P(n=6)}{P(n \mbox{ even})}$

CB

3. In a photoelectric detector, the number of photoelectrons
Y produced in time t depends on the (normalized) incident energy
X. If X were constant, say, X = x, Y would be a Poisson random variable with mean x, but as real light sources (except for gain-stabilized lasers) do not emit constant energy signals, X must be treated as a random variable. In certain situations X takes ¯nite number of values with probability mass function given by

P(X = xi) = pi; i = 1; : : : ; m:

Show that the average value of photoelectrons Y equals to the average value of the incident energy X, i.e.

E(Y ) = E(X):
$\displaystyle E(Y)=\sum_{i}E(Y|X=x_i)P(X=x_i)$

CB