Re: How to calculate the expectation of an exponantial function?

On Nov 10, 4:18 pm, Jack Tomsky
<jtom...@xxxxxxxxxxxxx> wrote:
Hello:

I would like to calcualate the expectation of an
exponantial function,
say f(x)=aexp(-bx). I know the equation of how to
calulate the
expectation, and I know it can be done somehow by
integral by parts to
get the final results, but even if I did the
integrate by parts, the
result is not so obvious. Does anyone know the
results of this
expectation in formular, or is there a place
lists
the table of this
so that I can directly cite it?

Thanks a lot for the help.

The expectation depends not only on the function of
x, which you gave as f(x) = a*exp(-bx), but also on
the distribution of x.  Specify the distribution of x
and I'll give you E(f(X)).

Jack

Thanks for the help, Jack. I don't know what the
distribution of x is.
Suppose it is normal distribution. When you specify
the equation, can
you please kindly explain a little bit why the
distribution of x is
needed? or kindly give a reference that I can read in
details about
this?

Thank you very much.


Expectation is a technical name for the average value. When you ask about the expectation of f(x), it depends on which values are x are most common or least common. The density function g(x) has an area under the curve of one and describes the relative probabilities of the various x values. (Actually, the probabilities of the intervals (x, x+d)),

In general, E(f(X)) is the integral of f(x) times g(x) dx.
In particular, let f(x)be the one you gave as a*exp(-bx). Suppose, as an example that X has an exponential distribution with the density function

g(x) = c*exp(-cx).

Then E(f(X)) = Integral[a*exp(-bx)*c*exp(-cx) dx] = ac*Integral[exp(-(b+c)x) dx] = ac/(b+c).

The integrals are both from zero to infinity.

As a reference, look up virtually any book on introduction to probability.

Jack
.

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