Computing MGF for Gamma Distribution

Let 𝑋 be a gamma variable with parameters 𝜆 and 𝛼 be defined as:

• 𝑓(𝑥) = (1/[𝛤(𝛼)𝜆^𝛼]) · 𝑥^𝛼-1· 𝑒^-𝑥/𝜆# for 𝑥 > 0

The moment generating function (MGF) of 𝑋 is defined as:

• $M_X(t)=(1-𝜆t{)}^{-𝛼}$

Computation Steps

Let 𝑋 be a gamma distribution with parameters 𝜆 and 𝛼. Thus:

• 𝑓(𝑥) = (1/[𝛤(𝛼)𝜆^𝛼]) · 𝑥^𝛼-1· 𝑒^-𝑥/𝜆# for 𝑥 > 0

where:

• 𝛼 - number of events
• 𝜆 - number of events per unit of time
• 𝛤(𝛼) = gamma function = (𝛼 - 1)!

Let’s compute the moment generating function (MGF) of 𝑋:

• $M_X(t)=\mathbf{E}[e^{tX}]$
• $M_X(t)=\mathbf{E}[e^{tX}]$
• 𝑓(𝑥) = (1/[𝛤(𝛼)𝜆^𝛼]) · 𝑥^𝛼-1· 𝑒^-𝑥/𝜆# for 𝑥 > 0
• $M_X(t)={\int }_0^{\infty }\frac{x^{𝛼-1}}{𝛤(𝛼)}\frac{e^{-x/𝜆}}{{𝜆}^{𝛼}}{e}^{tx}dx$
• $M_X(t)=\frac{1}{𝛤(𝛼){𝜆}^{𝛼}}{\int }_0^{\infty }{x}^{𝛼-1}{e}^{-x/𝜆}{e}^{tx}dx$
• $M_X(t)=\frac{1}{{𝜆}^{𝛼}}\frac{1}{𝛤(𝛼)}{\int }_0^{\infty }{x}^{𝛼-1}{e}^{-x(1/𝜆-t)}dx$
  • $x^{𝛼-1}{e}^{-x(1/𝜆-t)}\text{ looks like }Gamma(𝛼,\frac{1}{1/𝜆-t})$
  • We know:
• • • $1={\int }_0^{\infty }\frac{1}{𝛤(𝛼)}(1/𝜆-t{)}^{𝛼}{x}^{𝛼-1}{e}^{-x(1/𝜆-t)}dx$
    • $(1/𝜆-t{)}^{-𝛼}=\frac{1}{𝛤(𝛼)}{\int }_0^{\infty }{x}^{𝛼-1}{e}^{-x(1/𝜆-t)}dx$
• $M_X(t)=\frac{1}{{𝜆}^{𝛼}}(1/𝜆-t{)}^{-𝛼}$
• $M_X(t)=[1/𝜆{]}^{𝛼}[1/(1/𝜆-t){]}^{𝛼}$
• $M_X(t)=[1/(𝜆(1/𝜆-t)){]}^{𝛼}$
• $M_X(t)=(1-𝜆t{)}^{-𝛼}$

Using the MGF to Compute Moments

Expand ui

• $\mathbf{E}[X]=\mathbf{E}[X^1]$
• $\mathbf{E}[X]=\frac{d}{dt}[M_X(t)]{|}_{t=0}$
• $\mathbf{E}[X]=\frac{d}{dt}[(1-𝜆t{)}^{-𝛼}]{|}_{t=0}$
• $\mathbf{E}[X]=[-𝛼(1-𝜆t{)}^{-𝛼-1}\ast -𝜆]{|}_{t=0}$
• $\mathbf{E}[X]=[𝛼𝜆(1-𝜆t{)}^{-𝛼-1}]{|}_{t=0}$
• $\mathbf{E}[X]=𝛼𝜆(1-𝜆0{)}^{-𝛼-1}$
• $\mathbf{E}[X]=𝛼𝜆(1{)}^{-𝛼-1}$
• $\mathbf{E}[X]=𝛼𝜆$

Expand ui

• $\mathbf{E}[X^2]=\mathbf{E}[X^2]$
• $\mathbf{E}[X^2]=\frac{d^2}{d{t}^2}[M_X(t)]{|}_{t=0}$
• $\mathbf{E}[X^2]=\frac{d^2}{d{t}^2}[(1-𝜆t{)}^{-𝛼}]{|}_{t=0}$
• $\mathbf{E}[X^2]=\frac{d}{dt}[-𝛼(1-𝜆t{)}^{-𝛼-1}\ast -𝜆]{|}_{t=0}$
• $\mathbf{E}[X^2]=\frac{d}{dt}[𝛼𝜆(1-𝜆t{)}^{-𝛼-1}]{|}_{t=0}$
• $\mathbf{E}[X^2]=[𝛼𝜆(-𝛼-1)(1-𝜆t{)}^{-𝛼-2}\ast -𝜆]{|}_{t=0}$
• $\mathbf{E}[X^2]=[-𝛼𝜆(𝛼+1)(1-𝜆t{)}^{-𝛼-2}\ast -𝜆]{|}_{t=0}$
• $\mathbf{E}[X^2]=[𝛼𝜆𝜆(𝛼+1)(1-𝜆t{)}^{-𝛼-2}]{|}_{t=0}$
• $\mathbf{E}[X^2]=𝛼𝜆𝜆(𝛼+1)(1-𝜆0{)}^{-𝛼-2}$
• $\mathbf{E}[X^2]=𝛼𝜆𝜆(𝛼+1)(1{)}^{-𝛼-2}$
• $\mathbf{E}[X^2]={𝛼}^2{𝜆}^2+𝛼{𝜆}^2$

Computing Variance(𝑋)

Variance is defined as:

• 𝑉𝑎𝑟(𝑋) = 𝐄[𝑋²] - 𝐄[𝑋]²

Let’s use the MGF of a Bernoulli distribution to calculate the variance of 𝑋:

• 𝑉𝑎𝑟(𝑋) = 𝐄[𝑋²] - 𝐄[𝑋]²
• 𝑉𝑎𝑟(𝑋) = [𝛼²𝜆² + 𝛼𝜆²] - [𝛼𝜆]² # from above
• 𝑉𝑎𝑟(𝑋) = 𝛼²𝜆² + 𝛼𝜆² - 𝛼²𝜆²
• 𝑉𝑎𝑟(𝑋) = 𝛼𝜆²