Thread: 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!

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:



CB

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):

CB