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squenshl
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Homework Statement
\textbf{The Logarithmic Series Distribution}. We will examine the properties of a the Logarithmic Series Distribution. We will check that is is a probability function and compute a general term for factorial moments and, hence, compute its mean and variance. This distribution is related to the Poisson distribution and has been examined extensively in the following article: Consider the function ##f(p) = -\ln{(1-p})## for ##|p| < 1##.
Let's define ##X^{(k)} = X(X-1)(X-2)\ldots (X-k+1)## we know that the factorial moment of ##N## is $$E\left(X^{(k)}\right) = -\frac{(k-1)!}{\ln{(1-p)}}\left(\frac{p}{1-p}\right)^k, \quad k = 1,2,3,\ldots$$
show that $$E(X) = -\frac{1}{\ln{(1-p)}}\frac{p}{1-p}$$
and that $$\text{Var}(X) = -\frac{1}{\ln{(1-p)}}\frac{p}{(1-p)^2}\left(1+\frac{p}{\ln{(1-p)}}\right).$$
Homework Equations
The Attempt at a Solution
To calculate ##E(X)## we just set ##k = 1## in ##E\left(X^{(k)}\right)## to get
$$E(X) = -\frac{(1-1)!}{\ln{(1-p)}}\left(\frac{p}{1-p}\right)^1 = -\frac{1}{\ln{(1-p)}}\frac{p}{1-p}$$
as required. To calculate ##\text{Var}(X)## we first must find ##E(X^2)##. Apparently
$$E\left(X^2\right) = E[X(X - 1)] + E(X) = \frac{1}{-\ln(1 - p)} \frac{p}{(1 - p)^2}$$ but I can't seem to get this so I can't use the standard variance formula using expected values. I guess the question I'm asking is how do you calculate ##E[X(X - 1)]## everything else I can do. Please help.