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6. (425 / 625) The probability of exactly n successes in N trials for an event of probability p is, P(n) = ??????????????? N! / ((N - n)!n!) p^n (1 - p)^(N - n) (1) Consider a situation in which the probability p is small ( p << 1 ) and in which one is interested in the case n << N . (a) Using the result ln( 1 - p ) ? -p for p << 1 , show that ( 1 - p )^(N - n) ? e^(-Np) . (b) Show that ??????????????? N! / ((N - n)!) ? N^n for n << N . (c) Hence show that (1) reduces to, P(n) = ?^n / n! e^(-?) (2) where ? = Np is the mean number of events needed to achieve exactly n successes. The distribution (2) is called the Poisson distribution. (d) Show that (2) is properly normalized in the sense that ? (n=0 to N) P(n) = 1 . HINT: The sum can be extended to infinity to an excellent approximation since P(n) is negligibly small when n ? N * (e) Find ?n? for the Poisson distribution. (f) Find the variance of n for the Poisson distribution. *If N is large, P(n) becomes very small if n ? N because of the smallness of the factor p^n when p << 1 . Hence P(n) is indeed only appreciable when n << N .

          6. (425 / 625) The probability of exactly n successes in N trials for an event of probability p is,
P(n) = ??????????????? N! / ((N - n)!n!) p^n (1 - p)^(N - n) (1)
Consider a situation in which the probability p is small ( p << 1 ) and in which one is interested in the case n << N .
(a) Using the result ln( 1 - p ) ? -p for p << 1 , show that ( 1 - p )^(N - n) ? e^(-Np) .
(b) Show that ??????????????? N! / ((N - n)!) ? N^n for n << N .
(c) Hence show that (1) reduces to,
P(n) = ?^n / n! e^(-?) (2)
where ? = Np is the mean number of events needed to achieve exactly n successes. The distribution (2) is called the Poisson distribution.
(d) Show that (2) is properly normalized in the sense that ? (n=0 to N) P(n) = 1 .
HINT: The sum can be extended to infinity to an excellent approximation since P(n) is negligibly small when n ? N *
(e) Find ?n? for the Poisson distribution.
(f) Find the variance of n for the Poisson distribution.
*If N is large, P(n) becomes very small if n ? N because of the smallness of the factor p^n when p << 1 . Hence P(n) is indeed only appreciable when n << N .

6. (425 / 625) The probability of exactly n successes in N trials for an event of probability p is,
P(n) = ??????????????? N! / ((N - n)!n!) p^n (1 - p)^(N - n) (1)
Consider a situation in which the probability p is small ( p << 1 ) and in which one is interested in the case n << N .
(a) Using the result ln( 1 - p ) ? -p for p << 1 , show that ( 1 - p )^(N - n) ? e^(-Np) .
(b) Show that ??????????????? N! / ((N - n)!) ? N^n for n << N .
(c) Hence show that (1) reduces to,
P(n) = ?^n / n! e^(-?) (2)
where ? = Np is the mean number of events needed to achieve exactly n successes. The distribution (2) is called the Poisson distribution.
(d) Show that (2) is properly normalized in the sense that ? (n=0 to N) P(n) = 1 .
HINT: The sum can be extended to infinity to an excellent approximation since P(n) is negligibly small when n ? N *
(e) Find ?n? for the Poisson distribution.
(f) Find the variance of n for the Poisson distribution.
*If N is large, P(n) becomes very small if n ? N because of the smallness of the factor p^n when p << 1 . Hence P(n) is indeed only appreciable when n << N .

Added by Laura B.

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