Computing MGF for Bernoulli Distribution

The moment generating function (MGF) of a binomial variable 𝑋 with parameters 𝑝 and 𝑛 is defined as:

$M_{X} (t) = [1 - p + p e^{t}]^{n}$

Computation Steps

Let 𝑋 be a binomial distribution with parameters 𝑝 and 𝑛, where 𝑋 is equal to the sum of 𝑛 Bernoulli variables 𝑌. Thus:

𝑋 = 𝛴_{1≤𝑖≤𝑛}𝑌_𝑖

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

$M_{X} (t) = E [e^{tX}]$

$M_{X} (t) = E [e^{t \sum_{i = 1}^{n} Y_{i}}]$

$M_{X} (t) = E [e^{t Y_{1}} e^{t Y_{2}} ... e^{t Y_{n}}]$

$M_{X} (t) = E [e^{t Y_{1}}] E [e^{t Y_{2}}] ... E [e^{t Y_{n}}]$

Each 𝐄[𝑒^𝑡𝑌] is the moment-generating function of a Bernoulli variable, which is equal to [1 - 𝑝 + 𝑝𝑒^𝑡]:

$M_{X} (t) = [1 - p + p e^{t}] [1 - p + p e^{t}] ... [1 - p + p e^{t}]$

$M_{X} (t) = [1 - p + p e^{t}]^{n}$

Using the MGF to Compute Moments

Expand ui

$E [X] = E [X^{1}]$

$E [X] = \frac{d}{d t} [M_{X} (t)] ∣_{t = 0}$

$E [X] = \frac{d}{d t} [[1 - p + p e^{t})^{n}] ∣_{t = 0}$

$E [X] = [n (1 - p + p e^{t})^{(n - 1)} p e^{t}] ∣_{t = 0}$

$E [X] = n (1 - p + p e^{0})^{(n - 1)} p e^{0}$

$E [X] = n (1 - p + p)^{(n - 1)} p$

$E [X] = n 1^{(n - 1)} p$

$E [X] = n p$

Expand ui

$E [X^{2}] = E [X^{2}]$

$E [X^{2}] = \frac{d ^{2}}{d t ^{2}} [M_{X} (t)] ∣_{t = 0}$

$E [X^{2}] = \frac{d ^{2}}{d t ^{2}} [(1 - p + p e^{t})^{n}] ∣_{t = 0}$

$E [X^{2}] = \frac{d}{d t} [n (1 - p + p e^{t})^{(n - 1)} p e^{t}] ∣_{t = 0}$

$E [X^{2}] = [n (n - 1) (1 - p + p e^{t})^{(n - 2)} p e^{t} p e^{t} + n (1 - p + p e^{t})^{(n - 1)} p e^{t}] ∣_{t = 0}$

$E [X^{2}] = n (n - 1) (1 - p + p e^{0})^{(n - 2)} p e^{0} p e^{0} + n (1 - p + p e^{0})^{(n - 1)} p e^{0}$

$E [X^{2}] = n (n - 1) (1 - p + p)^{(n - 2)} pp + n (1 - p + p)^{(n - 1)} p$

$E [X^{2}] = n (n - 1) (1)^{(n - 2)} pp + n (1)^{(n - 1)} p$

$E [X^{2}] = n (n - 1) pp + n p$

$E [X^{2}] = (n^{2} - n) p^{2} + n p$

$E [X^{2}] = n^{2} p^{2} - n p^{2} + n p$

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) = [𝑛²𝑝² - 𝑛𝑝² + 𝑛𝑝] - [𝑛𝑝]² # from above

𝑉𝑎𝑟(𝑋) = 𝑛²𝑝² - 𝑛𝑝² + 𝑛𝑝 - 𝑛²𝑝²

𝑉𝑎𝑟(𝑋) = 𝑛𝑝 - 𝑛𝑝²

𝑉𝑎𝑟(𝑋) = 𝑛(𝑝 - 𝑝²)

𝑉𝑎𝑟(𝑋) = 𝑛𝑝(1 - 𝑝)

／var／log marcus chiu

Explorer

Moment-Generating Function - Binomial Distribution

Computing MGF for Bernoulli Distribution

Using the MGF to Compute Moments

／var／logmarcus chiu

Explorer

Moment-Generating Function - Binomial Distribution

Computing MGF for Bernoulli Distribution

Using the MGF to Compute Moments

／var／log marcus chiu