Computing MGF for Geometric Distribution

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

• $M_X(t)=\frac{p{e}^t}{1-q{e}^t}$

Computation Steps

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

• 𝐏(𝑋=𝑥) = (1−𝑝)^𝑥−1𝑝

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

• $M_X(t)=E[e^{tX}]$
• $M_X(t)={\sum }_{i=1}^{\infty }{e}^{t\cdot x_i}P(X=x_i)$
• $M_X(t)={\sum }_{i=1}^{\infty }{e}^{t\cdot (x_i-1)}{e}^{t}p(1-p{)}^{x_i-1}$
• $M_X(t)=e^{t}p{\sum }_{i=1}^{\infty }{e}^{t\cdot (x_i-1)}(1-p{)}^{x_i-1}$
• $M_X(t)=e^{t}p{\sum }_{i=1}^{\infty }[(1-p)e^t{]}^{x_i-1}$
• $M_X(t)=e^{t}p{\sum }_{i=1}^{\infty }[q{e}^t{]}^{x_i-1}$
• $M_X(t)=\frac{p{e}^t}{1-q{e}^t}$

Using the MGF to Compute Moments

TODO