- Logmoment generating function
In
mathematics , the logarithmic momentum generating function (equivalent tocumulant 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
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