homework1 - P(n = N n(N" n p n(1" p N" n b Now...

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PHYSICS 112 S. G. LOUIE FALL 2006 Problem Set #1 Due: 09/08/06 Reading : Chapters 1 and 2 of Kittel and Kroemer (20) 1. Consider the random walk problem for a particle in one dimension. Assume that in each step its displacement is always positive and equally likely to be anywhere in the range between l " b and l + b where b < l . After N steps, what is: a) the mean displacement x ? b) the dispersion < (x " x ) 2 > ? (Suggestion: use the central limit theorem.) (20) 2. a) Show that the probablility P(n) that an event characterized by a probability p occurs n time in N trials is given by the binomial distribution
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Unformatted text preview: P(n) = N! n!(N " n)! p n (1 " p) N " n . b) Now consider a situation where p << 1 and where one is interested in the case n << N. (This may be the case of emission of α particles by a radioactive source.) Show that the above expression in a) becomes P(n) = " n n! e #" where λ = Np is the mean number of events. This is called the Poisson distribution . (The following approximations will be useful: (1 " p) N " n # e " Np and N! /(N " n)! # N n ). (20) 3. Problem 1, p. 52 of K&K. (20) 4. Problem 2, p. 52 of K&K. (20) 5. Problem 4, p. 53 of K&K....
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