Moment Generating Function of Binomial Distribution

Author: Bindeshwar Singh Kushwaha
Institute: PostNetwork Academy

Definition of Moment Generating Function

• The moment generating function (m.g.f.) of a random variable \( X \) is defined as:
  \[
  M_X(t) = E(e^{tX})
  \]
• For a continuous random variable:
  \[
  M_X(t) = \int_{-\infty}^{\infty} e^{tx} f(x) \, dx
  \]
• For a discrete random variable:
  \[
  M_X(t) = \sum_x e^{tx} f(x)
  \]

Series Expansion of MGF

• Using Taylor series expansion of \( e^{tX} \):
  \[
  e^{tX} = 1 + tX + \frac{t^2 X^2}{2!} + \frac{t^3 X^3}{3!} + \cdots + \frac{t^r X^r}{r!} + \cdots
  \]
• Taking expectation:
  \[
  M_X(t) = 1 + t E(X) + \frac{t^2}{2!} E(X^2) + \cdots + \frac{t^r}{r!} E(X^r) + \cdots
  \]
• This is:
  \[
  M_X(t) = \sum_{r=0}^\infty \frac{t^r}{r!} \mu_r’
  \]
  where \( \mu_r’ = E(X^r) \) is the \( r \)th moment about origin.

Generating Moments Using MGF

• Since MGF generates moments, it is called the moment generating function.
• Differentiating \( M_X(t) \) \( r \) times w.r.t. \( t \), we get:
  \[
  \left. \frac{d^r}{dt^r} M_X(t) \right|_{t=0} = \mu_r’
  \]
• Example:
  \[
  \frac{d}{dt} M_X(t) = E\left[ X e^{tX} \right], \quad \text{and at } t = 0, \quad E(X)
  \]

MGF of Binomial Distribution

• Let \( X \sim \text{Bin}(n, p) \), then:
  \[
  M_X(t) = E(e^{tX}) = \sum_{x=0}^n e^{tx} \binom{n}{x} p^x q^{n-x}
  \]
• Rewriting:
  \[
  = \sum_{x=0}^n \binom{n}{x} (qe^t)^x (pe^t)^{n-x}
  \]
• Combine:
  \[
  = (q e^t + p e^t)^n
  \]
• Let \( q = 1 – p \), then:
  \[
  M_X(t) = (q + p e^t)^n
  \]
• Now calculate moments:
• Mean (\( \mu \)):
  \[
  \mu = M_X'(0) = np
  \]
• Variance (\( \sigma^2 \)):
  \[
  \sigma^2 = M_X”(0) – (M_X'(0))^2 = npq
  \]

mgfbinomial2

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