Review:

Moment Generating Functions

Learning Outcomes

• Define Moment Generating Functions

• Discuss Properties

Moment Generating Functions

• Moment Generating Functions

• Poisson Distribution

• Binomial Distribution

• Uniform Distribution

• Normal Distribution

• MGF Properties

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

\[ m(t) = E(e^{tX}) \]

The \(k\)th moment can be obtained by taking the \(k\)th derivative of the mgf, with respect to \(t\), and setting \(t\) equal to 0:

\[ E(X^k)=\frac{d^km(t)}{dt}\Bigg|_{t=0} \]

Poisson Distribution

• Moment Generating Functions

• Poisson Distribution

• Binomial Distribution

• Uniform Distribution

• Normal Distribution

• MGF Properties

MGF

Expected Value

Variance

Variance

Binomial Distribution

• Moment Generating Functions

• Poisson Distribution

• Binomial Distribution

• Uniform Distribution

• Normal Distribution

• MGF Properties

MGF

Uniform Distribution

• Moment Generating Functions

• Poisson Distribution

• Binomial Distribution

• Uniform Distribution

• Normal Distribution

• MGF Properties

MGF

Normal Distribution

• Moment Generating Functions

• Poisson Distribution

• Binomial Distribution

• Uniform Distribution

• Normal Distribution

• MGF Properties

MGF

MGF Properties

• Moment Generating Functions

• Poisson Distribution

• Binomial Distribution

• Uniform Distribution

• Normal Distribution

• 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

\[ M_Y(t) = e^{tb}M_X(at) \]

Derivation

Linearity

Let \(X\) and \(Y\) be two random variables with MGFs \(M_X(t)\) and \(M_Y(t)\), respectively, and are independent. The MGF of \(U=X-Y\)

\[ M_U(t) = M_X(t)M_Y(-t) \]

Derivation

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

Uniqueness

Let \(X_1,\cdots, X_n\) be independent random variables, where \(X_i\sim N(\mu_i, \sigma^2_i)\), with \(M_{X_i}(t)=\exp\{\mu_i t+\sigma^2_it^2/2\}\) for \(i=1,\cdots, n\). Find the MGF of \(Y=a_1X_1+\cdots+a_nX_n\), where \(a_1, \cdots, a_n\) are constants.