[Verse 1]
Bernoulli flips a single coin tonight
P of one equals p, zero gets one minus p
Expected value p, variance p times its complement
One trial tells the tale, success or fail
[Chorus]
Seven distributions dancing through the math
Bernoulli, Binomial, Poisson on the path
Geometric waits, Normal curves so smooth
Exponential decays, Uniform stays aloof
Learn their formulas, know when each applies
Probability's cast of characters in disguise
[Verse 2]
Binomial counts successes in n tries
C-n-k times p to k times complement to n minus k
Expected np, variance adds the complement factor
Multiple trials, counting up the wins
[Chorus]
Seven distributions dancing through the math
Bernoulli, Binomial, Poisson on the path
Geometric waits, Normal curves so smooth
Exponential decays, Uniform stays aloof
Learn their formulas, know when each applies
Probability's cast of characters in disguise
[Verse 3]
Poisson models rare events that strike
E to minus lambda, lambda to k, divide by k factorial
Expected lambda, variance also lambda too
Rare occurrences scattered through time
[Bridge]
Geometric waits for first success to show
One minus p to k minus one times p
Expected one over p attempts until we win
Normal bell curve, mu and sigma squared
One over sigma root two pi, e to minus x minus mu squared over two sigma squared
Expected mu, variance sigma squared
[Verse 4]
Exponential measures waiting time decay
Lambda e to minus lambda x for x greater equal zero
Expected one over lambda, variance one over lambda squared
Uniform spreads evenly from a up to b
One over b minus a across the interval
Expected a plus b over two, variance b minus a squared over twelve
[Outro]
Cast of characters, each one has its role
Recognize the pattern, probability's goal
From coin flips to decay, from rare to routine
These seven distributions rule the random scene