Statistical distributions

Discrete variables with two outcomes such as success/failure or coin flips.

For a single trial, the Bernoulli distribution:

\[ P(X = x) = f(x;p) = p^x (1-p)^{1-x}, \]

where \(p\) is the probability that \(x = 1\) and \(1 - p\) is the probability that \(x = 0\).

Properties:

For \(x\) successes out of \(n\) trials, the Binomial distribution:

\[ P(X = x) = f(x; n, p) = \binom{n}{x} p^x (1-p)^{n-x}, \]

where \(n\) is the number of trials and \(x\) is the number of successes.

Properties:

Time between independent events with a consistent average rate, the exponential distribution:

\[ P(X = x) = f(x; \lambda) = \lambda e^{- \lambda x}, \]

where \(\lambda\) is the rate, and \(x\) is the time between events.

Properties:

For given number of independent events with a consistent average rate occurring in a fixed time, the Poisson distribution:

\[ P(X = x) = f(x; \lambda) = \frac{\lambda^x e^{- \lambda}}{x!}, \]

where \(\lambda\) is the rate, and \(x\) is the number of events.

Properties:

\(E(X) = Var(X) = \lambda\)
if \(n\) is large and \(p\) is small, a Poisson with \(\lambda = n p\) approximates a binomial

Continuous real-valued variables, the normal distribution:

\[ P(X = x) = f(x; \mu, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}} e ^{-\frac{1}{2} (\frac{x - \mu}{\sigma})^2}, \]

Properties:

\(E(X) = \mu\)
\(Var(X) = \sigma^2\)
1, 2, 3, standard deviations contain 68% 95% 99% of the density
the transformation \(Z = \frac{X - \mu}{\sigma}\) gives \(\mu = 0\) and \(\sigma = 1\)
with \(\mu = 0\) and \(\sigma = 1\), it’s called a standard normal

\[ P(Z = x) = f(x) = \frac{1}{\sqrt{2 \pi}} e ^{- \frac{x^2}{2}}, \]