Logmoment generating function

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

:$mu\_\{Y\}(s)=ln\; E(e^\{scdot\; Y\})$

where "Y" is a random variable.

Thus, if "Y" is a discrete random variable, then

:$mu\_\{Y\}(s):=ln\; sum\_y\; P(y)cdot\; e^\{scdot\; y\}\; ,$

especially for the binary case (Bernoulli distribution)

:$mu\_Y(s)=lnleft\{pcdot\; e^s\; +\; (1-p)\; ight\}$

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

:$mu\_\{Y\}(s):=ln\; int\_y\; Phi(y)cdot\; e^\{scdot\; y\}.$

Here Φ 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:

:$mu\_Y(s)=ln\; left(e^\{scdot\; Y\}\; ight)\; =\; ln\; Eleft(e^\{scdot\; sum\_\{j=1\}^J\; X\_j\}\; ight)stackrel\{*\}\{=\}\; lnprod\_\{j=1\}^\{J\}\; Eleft(e^\{scdot\; X\_j\}\; ight)\; =sum\_\{j=1\}^J\; ln\; Eleft(e^\{scdot\; X\_j\}\; ight)\; =\; sum\_\{j=1\}^J\; mu\_\{X\_j\}(s).$

("*" is where we used the independence of the $X\_j$ random variables)

ee also

*Cumulant generating function
*Moment generating function