Statistics: Binomial, Normal & Other Distributions

BINS: - Binary: each trial is success or failure - Independent trials - Number of trials n is fixed - Same probability of success p every trial

P(X = k) = \dbinom{n}{k} p^k (1 - p)^{n - k}

\mu = np , \sigma = \sqrt{np(1 - p)}

3/8

Counts trials until the first success: P(X = k) = (1 - p)^{k-1} p , with mean \dfrac{1}{p}

Counts events in a fixed interval: P(X = k) = \dfrac{\lambda^k e^{-\lambda}}{k!} , with mean = variance = \lambda

Symmetric and bell-shaped; mean = median = mode; fully described by \mu and \sigma ; written N(\mu, \sigma) .

The normal distribution with mean 0 and standard deviation 1, Z \sim N(0, 1)

0.5

1.645 , 1.96 and 2.576

0. Probabilities for continuous variables are areas under the density curve, and a single point has no area.

PMF: probability of each value of a discrete variable. PDF: density curve of a continuous variable; area gives probability. CDF: F(x) = P(X ≤ x), for either type.

Mean \dfrac{a + b}{2} , variance \dfrac{(b - a)^2}{12}

Models waiting time between events at rate \lambda ; mean \dfrac{1}{\lambda} . It is memoryless: how long you have already waited does not change the future wait.

For large n , the sampling distribution of \bar{x} is approximately normal with mean \mu and standard deviation \dfrac{\sigma}{\sqrt{n}} , whatever the shape of the population.

SE_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}

Mean p , standard deviation \sqrt{\dfrac{p(1 - p)}{n}}

When np \ge 10 and n(1 - p) \ge 10 (some textbooks use 5).

Both are symmetric and centered at 0, but t has heavier tails. It has n - 1 degrees of freedom for one sample and approaches the normal as df grows.

Takes only nonnegative values, is skewed right, and is defined by its degrees of freedom. Used for goodness-of-fit, independence and homogeneity tests.

Discrete: countable values (number of heads, number of customers). Continuous: any value in an interval (height, time).