## PROBLEMS

**3.1** Given the Poisson random variable with the discrete probability in Eq. (3.3.19), its *j*th moment is computed as

Determine the mean value, mean-square value, and variance of the Poisson variable.

**3.2** The characteristic function Ψ(*j*ω) of any discrete random variable is computed from its probability function *P*(*k*) as

The moment generating function *M*(*z*) is obtained from Ψ(*j*ω) by

replacing *j*ω = in(1 − *z*). Show that

**3.3** Use the results of Problem 3.2 to derive the Poisson count probability from its characteristic function.

**3.4** Electrons are emitted from a photodetecting surface according to a Poisson probability with mean value *m*. Suppose, however, the probability that a given emitted electron will be collected at the output is η. Determine the resulting count probability of the electrons collected at the output. *Hint*: Treat the conversion of each emitted electron to an output electron as a binary random variable with probability η.

**3.5** Let *W _{n}* be the time for

*n*events to occur in a counting process

*k*(

*t*). The variable

*W*is called the

_{n}*waiting time*to the

*n*th event. Show that if the events occur with a Poisson probability with mean rate ν events/second then

- (a)
- (b)
- (c)
- (d) ...

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