Lectures on Elementary Probability

6.7. SUMMARY 43
Theorem 6.11 The expectation of the gamma random variable is
Proof: Compute the integral
E[T r ] =
E[T r ] = r 1 λ . (6.41)
∫ ∞
0
t (λt)(r−1)
(r − 1)! e−λt λ dt. (6.42)
After a change of variable it becomes a Γ(r + 1) integral.
Alternative proof: Use the formula for the mean of an exponential random
variable and the fact that the mean of a sum is the sum of the means.
Theorem 6.12 The variance of the gamma random variable is
Proof: Compute the integral
E[T 2 r ] =
∫ ∞
0
σ 2 T r
= r 1 λ 2 . (6.43)
t 2 (λt)(r−1)
(r − 1)! e−λt λ dt. (6.44)
After a change of variable it becomes a Γ(r+2) integral. The result is (r+1)r/λ 2 .
It follows that Var(T r ) = E[T 2 r ] − E[T r ] 2 = r/λ 2 .
Alternative proof: Use the formula for the variance of an exponential random
variable and the fact that the variance of a sum of independent random variables
is the sum of the variances.
These formula for the mean and expectation of the gamma waiting time are
quite important. When r = 1 this is the exponential waiting time, so we already
know the facts. So consider the case when we are waiting for the rth success and
r is large. The formula for the mean is r/λ, which is a quite intuitive result. The
formula for the variance is also important. It is better to look at the standard
deviation. This is √ r/λ. Note that the standard deviation is considerably
smaller than the mean. This means that the distribution is somewhat peaked
about the mean, at least in a relative sense. Very small values are unlikely, so
are very large values (relative to the mean). The reason for this behavior is that
the waiting time for the rth success is the sum of r independent exponential
waiting times. These are individually quite variable. But the short exponential
waiting times and the long exponential waiting times tend to cancel out each
other. The waiting time for the rth success is comparatively stable.
6.7 Summary
Consider a Poisson process with rate λ. Thus the expected number of jumps in
an interval of time of length t is λt. The random variable N(t) that counts the
number of jumps up to and including time t is a Poisson random variable. The