Week 6
Learning Outcomes
First Lecture
Moment Generating Functions
MGF Properties
Second Lecture
- R Lab Continuous Random Variables
Lecture
Videos
| Section | Tuesday | Thursday |
|---|---|---|
| 001 | Video | Video |
| 002 | Video | Video |
Important Concepts
First Lecture
Moments
The \(k\)th moment is defined as the expectation of the random variable, raised to the \(k\)th power, defined as \(E(X^k)\).
Moment Generating Functions
The moment generating functions is used to obtain the \(k\)th moment. The mgf is defined as
\[ m(c) = E(e^{tX}) \]
The \(k\)th moment can be obtained by taking the \(k\)th derivative of the mgf, with respect to \(c\), and setting \(c\) equal to 0:
\[ E(X^k)=\frac{d^km(c)}{dc}\Bigg|_{c=0} \]
MGF Properties
Linearity
Let \(X\) follow a distribution \(f\), with the an MGF \(M_X(t)\), the MGF of \(Y=aX+b\) is given as
\[ M_Y(t) = e^{tb}M_X(at) \]
Let \(X\) and \(Y\) be two random variables with MGFs \(M_X(t)\) and \(M_Y(t)\), respectively, and are independent. The MGF of \(U=X-Y\)
\[ M_U(t) = M_X(t)M_Y(-t) \]
Uniqueness
Let \(X\) and \(Y\) have the following distributions \(F_X(x)\) and \(F_Y(y)\) and MGFs \(M_X(t)\) and \(M_Y(t)\), respectively. \(X\) and \(Y\) have the same distribution \(F_X(x)=F_Y(y)\) if and only if \(M_X(t)=M_Y(t)\).