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The moment generating function (MGF) is a powerful tool that encodes all the moments of a distribution in a single function. It can be used to find means and variances, prove distributional results, and identify distributions. This topic is part of the AQA Further Mathematics specification.
The moment generating function of a random variable X is defined as:
MX(t)=E(etX)
For a discrete random variable:
MX(t)=∑xetxP(X=x)
For a continuous random variable:
MX(t)=∫−∞∞etxf(x)dx
The MGF exists (is finite) for all t in some interval containing 0.
Expanding etX as a Taylor series:
etX=1+tX+2!(tX)2+3!(tX)3+⋯
Taking expectations:
MX(t)=1+tE(X)+2!t2E(X2)+3!t3E(X3)+⋯
So the n-th moment is obtained by differentiating n times and evaluating at t=0:
E(Xn)=MX(n)(0)
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