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¶

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