Homework PH353 2. Due Monday, April 17 I. The Poisson distribution is useful for events that have a very small probability of success, like the random decay of atomic nuclei or emission of a photon from a very dim light source. It can be derived by considering the toss of a very unfair coin: p(heads) << 1. The exact formula for the probability of obtaining exactly n heads in N tosses of an unfair coin is: N! P ( N , n) = p n (1 - p ) N - n ( N - n)!n ! You should be able to derive this! As shown in class, for the special case of p<<1, this can be simplified to n! Where = Np is the mean or expected number of (in events this case heads) in N trials. P(N,n) is called the Poisson distribution. (a) Prove by direct summation that the mean number of events is:
< n >= nP ( N , n) =
n=0
P ( N , n)
n
e-
(b) Prove by direct summation that the variance, or mean square deviation from the mean, is:
2 =< (n - < n >) 2 >=< n 2 > - < n > 2 = (n - n ) 2 ...