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The transform associated with a ramdom variable \(X\) is defined by
\[M_X(s) = E[e^{sX}],\]where \(s\) is a scalar parameter.
Transform is also referred to moment generating function.
When \(X\) is discrete, the corresponding transform is given by
\[M(s) = \sum_xe^{sx}p_X(x),\]while in the continous case it is given by
\[M(s) = \int_{-\infty}^{\infty}e^{sz}f_X(x)dx.\]下標 \(X\) 在上下文明確時可省略。
Given a transform \(M_X(s)\), the distribution of \(X\) can be uniquely determined.
也就是說，可以從 transform 來倒推 r.v. 的 CDF！
In fact, more mathematical condition are required, but it suffices to know this simplified fact first.
Let \(Z=X+Y\), and \(X\) and \(Y\) are independent r.v.s, and by definition
\[M_Z(s) = E[e^sZ] = E[e^s(X+Y)] = E[e^{sX}e^{sY}].\]Since \(X\) and \(Y\) are independent, we have
\[M_Z(s) = E[e^{sX}]E[e^{sY}] = M_X(s)M_Y(s).\]Recall that the multiplication of two generating functions is associated with convolution, which satisfies the result here.
By definition, \(M_X(s) = E[e^{sX}]\). If we differentiate it, we get
\[{d\over ds}M_X(s) = {d\over ds}E[e^{sX}] = E\Big[{d\over ds}e^{sX}\Big] = E[Xe^{sX}],\]which means that
\[{d\over ds}M_X(s)\Bigg\vert_{s=0} = E[X].\]將 \(E[\cdot]\) 依定義展開就可以看出為何 \({d\over ds}\) 可以任意移動。
We can further derive that
\[{d^n\over ds^n}M_X(s)\Bigg\vert_{s=0} = E[X^n].\]只要有 transform，求 moment 變得輕鬆！
Particularly, for standard normal, we have
\[M(s) = e^{s^2/2}\]