Binomial and Geometric Distributions¶

Binomial Distributions¶

Counts the number of success

\(X \sim B(n, p)\)

• Fixed number of repeated trials
• Trials are independent
• Two outcomes
• Probability of success is constant

Shape of Binomial Distributions¶

\(p\) in \(X \sim B(n, p)\) determines the skewness

\(p\)	Skewness
\(< 0.5\)	Positive
\(= 0.5\)	Symmetrical
\(< 0.5\)	Negative

Mean & Standard Deviation of Binomial Distributions¶

\(\mu_X = np\)

\(\sigma_X = \sqrt{np(1-p)}\)

Probabilities for Binomial Distributions¶

\(P(X=x) = C_n^x \cdot p^x (1-p)^{n-x}\)

Geometric Distributions¶

Counts the number of trials needed to obtain the first success

\(X \sim Geo(P)\)

• Trials are independent
• Two outcomes
• Probability of success is constant

Shape of Geometric Distributions¶

• Geometric distributions always have positive skew

Mean & Standard Deviation of Geometric Distributions¶

\(\mu_X = \dfrac1p\)

\(\sigma_X = \dfrac{\sqrt{1-p}}p\)

Probabilities for Geometric Distributions¶

\(P(X=x) = (1-p)^{x-1} p\)