Your questions does not make much sense.
In your example, you have a random variable X with an exponential distribution. The probability density function (pdf) of the exponential distribution with parameter [lambda] is p(x ; [lambda]) := [lambda]*e(-[lambda]*x) for x >= 0. The expected value (mean) of an exponentially distributed random variable is 1/[lambda].
Now, your question mentions a function p(x; [alpha], [beta]) := [alpha]*exp(-[beta] *x ), which is not clearly a pdf. For example, it needs to integrate to 1 so you need to know something else about the parameters [alpha] and [beta]. I am going to take a guess and say that you are interested in a Gamma distribution, whose mean can be found here:
http://en.wikipedia.org/wiki/Gamma_distribution
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