# Week 6

## Learning Outcomes

### First Lecture

Moment Generating Functions

MGF Properties

### Second Lecture

- R Lab Continuous Random Variables

## 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 \(M_X(t)=M_Y(t)\).

## Resources

### First Lecture

Slides | Videos |
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Slides | Video 001 Video 002 |

### Second Lecture

Lab | Videos |
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Lab |