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)

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

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}right) = ln Eleft(e^{scdot sum_{j=1}^J X_j}right) stackrel{*}{=} lnprod_{j=1}^{J} Eleft(e^{scdot X_j}right) = sum_{j=1}^J ln Eleft(e^{scdot X_j}right) = sum_{j=1}^J mu_{X_j}(s).

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

See also

• Cumulant generating function
• Moment generating function