Sometimes the poisson distribution, described in the previous section, is called an exponential distribution. After all, it is based on an exponential curve. However, in other contexts, the phrase "exponential distribution" means something quite different. With t as a variable and p as a fixed parameter, the exponential distribution has the following density function.
p2tE-pt
This curve starts at the origin, attains a maximum at the point 1/p,pE, then approaches 0 as t approaches infinity. Integrate to obtain the following distribution function, which starts at zero and approaches 1 as t approaches infinity.
1 - (1+pt)E-pt
Multiply the density function by t and integrate to obtain the mean. Multiply the density function by t2 and integrate, then subtract the mean squared, to obtain the variance. This is all integration by parts, and I'll leave it to you. The mean is 2/p and the standard deviation is sqrt(2)/p.