Logmoment generating function

In mathematics, the logarithmic momentum generating function (equivalent to cumulant generating function) (logmoment gen func) is defined as follows:


 * $$\mu_{Y}(s)=\ln E(e^{s\cdot Y})$$

where Y is a random variable.

Thus, if Y is a discrete random variable, then


 * $$\mu_{Y}(s):=\ln \sum_y P(y)\cdot e^{s\cdot y} ,$$

especially for the binary case (Bernoulli distribution)


 * $$\mu_Y(s)=\ln\left\{p\cdot e^s + (1-p)\right\}$$

and if Y is a random variable with continuous distribution, then


 * $$\mu_{Y}(s):=\ln \int_y \Phi(y)\cdot e^{s\cdot y}.$$

Here &Phi; is the cumulative distribution function of Y.

it is also true that for a sum of independent random variables


 * $$Y=\sum_{j=1}^J X_j$$

that


 * $$\mu_Y(s)=\sum_{j=1}^J \mu_{X_j}(s)$$

Proof:



\mu_Y(s)=\ln \left(e^{s\cdot Y}\right) = \ln E\left(e^{s\cdot \sum_{j=1}^J X_j}\right) \stackrel{*}{=} \ln\prod_{j=1}^{J} E\left(e^{s\cdot X_j}\right) = \sum_{j=1}^J \ln E\left(e^{s\cdot X_j}\right) = \sum_{j=1}^J \mu_{X_j}(s). $$

("*" is where we used the independence of the $$X_j$$ random variables)