Geometric Distribution

Probability Mass Function

• 𝐏(𝑋=𝑥) = 𝐏{the 1^𝑠𝑡 success occurs on the 𝑥^𝑡ℎ bernoulli trial}
• 𝐏(𝑋=𝑥) = (1−𝑝)^𝑥−1𝑝

Expectation

𝐄[𝑋] = 1/𝑝

Click here to expand...

• 𝐄[𝑋] = 𝛴_{1≤𝑥≤∞}[𝑥·𝐏(𝑋=𝑥)] # see calculating the expected value of a discrete distribution
• 𝐄[𝑋] = 𝛴_{1≤𝑥≤∞}[𝑥·(1-𝑝)^𝑥−1𝑝]
• 𝐄[𝑋] = 𝛴_{1≤𝑥≤∞}[𝑥𝑞^𝑥−1𝑝] # let 𝑞 = 1-𝑝
• 𝐄[𝑋] = 𝑝 * 𝛴_{1≤𝑥≤∞}[𝑥𝑞^𝑥−1]

• 𝐄[𝑋] = 𝑝 * [1/(1−𝑞)²] # see expand below

Click here to expand... geometric series (infinite) 𝑠(𝑞) = 𝛴_{0≤𝑥≤∞}[𝑞^𝑥] equals 1/(1-𝑞) for 𝑞<1

taking the derivative of 𝑞:

List indent undo

• 𝐄[𝑋] = 𝑝 * [1/(1−(1-𝑝))²]
• 𝐄[𝑋] = 𝑝 * [1/(1−1+𝑝)²]
• 𝐄[𝑋] = 𝑝 * [1/(0+𝑝)²]
• 𝐄[𝑋] = 𝑝 * (1/𝑝²)
• 𝐄[𝑋] = 𝑝/𝑝²
• 𝐄[𝑋] = 1/𝑝

Variance

𝑉𝑎𝑟(𝑋) = (1-𝑝) / 𝑝²

Click here to expand...

proof of variance https://math.stackexchange.com/questions/1299465/proof-variance-of-geometric-distribution

Cumulative Distribution Function

TODO - http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Geometric.pdf

Moment Generating Function

• $M-X(t)=E[e^{tX}]=\frac{p{e}^t}{1-q{e}^t}$

See: Moment-Generating Function - Geometric Distribution

Subpages

• Geometric Distribution vs Binomial Distribution
• Geometric Distribution vs Negative Binomial Distribution
• Geometric Distribution is the only Discrete Distribution with the Memoryless Property (similar to its continuous counterpart Exponential Distribution)