Moment Generating Functions

Moment Generating Function

For a random variable $X$, the moment generating function is defined as:

$$M_X(t)=E(e^{tx})$$

The $r^{\text{th}}$ raw moment can be obtained from the moment generating function as follows:

$$M_r^{\prime }=\frac{d^r}{d{t}^r}{M}_X(t){|}_{t=0}$$ $$M_r^{\prime }=\text{coefficient of}\frac{t^r}{r!}\text{in the Taylor series of}{M}_X(t)$$

References

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• date: 2025.03.16
• time: 20:42