Computing MGF for Bernoulli Distribution

The moment generating function (MGF) of a Bernoulli variable 𝑋 with parameter 𝑝 is defined as:

• 𝑀_𝑋(𝑡) = 1-𝑝 + 𝑒^𝑡𝑝

Computation Steps

Let 𝑋 be a Bernoulli distribution with parameter 𝑝. Thus:

• 𝑃(𝑋=1) = 𝑝
• 𝑃(𝑋=0) = 1-𝑝

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

• 𝑀_𝑋(𝑡) = 𝐄[𝑒^𝑡𝑋]
• 𝑀_𝑋(𝑡) = 𝛴_𝑥∊𝑋[𝑒^𝑡𝑥·𝑃(𝑋=𝑥)]
• 𝑀_𝑋(𝑡) = 𝑒^𝑡0·𝑃(𝑋=0) + 𝑒^𝑡1·𝑃(𝑋=1)
• 𝑀_𝑋(𝑡) = 𝑃(𝑋=0) + 𝑒^𝑡·𝑃(𝑋=1)
• 𝑀_𝑋(𝑡) = 1-𝑝 + 𝑒^𝑡𝑝

Using the MGF to Compute Moments

Expand ui

• $\mathbf{E}[X]=\mathbf{E}[X^1]$
• $\mathbf{E}[X]=\frac{d}{dt}[M_X(t)]{|}_{t=0}$
• $\mathbf{E}[X]=\frac{d}{dt}[1-p+e^{t}p]{|}_{t=0}$
• $\mathbf{E}[X]=[0-0+e^{t}p]{|}_{t=0}$
• $\mathbf{E}[X]=p$

Expand ui

• $\mathbf{E}[X^2]=\mathbf{E}[X^2]$
• $\mathbf{E}[X^2]=\frac{d^2}{d{t}^2}[M_X(t)]{|}_{t=0}$
• $\mathbf{E}[X^2]=\frac{d^2}{d{t}^2}[1-p+e^{t}p]{|}_{t=0}$
• $\mathbf{E}[X^2]=\frac{d}{dt}[0-0+e^{t}p]{|}_{t=0}$
• $\mathbf{E}[X^2]=[e^{t}p]{|}_{t=0}$
• $\mathbf{E}[X^2]=p$

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) = 𝑝 - 𝑝² # from above
• 𝑉𝑎𝑟(𝑋) = 𝑝(1-𝑝)