Introductory Statistics: Probability Distributions and Discrete Distributions

Make a frequency distribution, sum frequencies, find probabilities by dividing each frequency by the total, and verify probabilities sum to 1 and are between 0 and 1.

The mean \(\mu\) is the sum of each value multiplied by its probability: \(\mu = \sum xP(x)\).

Variance \(\sigma^2\) is \(\sigma^2 = \sum (x - \mu)^2 P(x)\).

Standard deviation \(\sigma\) is the square root of the variance: \(\sigma = \sqrt{\sigma^2} = \sqrt{\sum (x - \mu)^2 P(x)}\).

The expected value is equal to the mean: \(E(x) = \mu = \sum xP(x)\).

1) Fixed number of independent trials.
2) Two possible outcomes per trial (success or failure).
3) Constant probability of success \(p\).
4) Random variable \(x\) counts number of successes.

\(P(x) = \binom{n}{x} p^x q^{n-x}\), where \(q = 1 - p\).

The mean is \(\mu = np\), where \(n\) is trials and \(p\) is success probability.

Variance is \(\sigma^2 = npq\), where \(q = 1 - p\).

Standard deviation is \(\sigma = \sqrt{npq}\).

A geometric distribution models the number of trials until the first success, with independent trials and constant success probability \(p\).

\(P(x) = p q^{x-1}\), where \(q = 1 - p\) and \(x\) is the trial of first success.

The Poisson distribution models the number of times an event occurs in a fixed interval, with independent occurrences and constant average rate \(\mu\).

\(P(x) = \frac{\mu^x e^{-\mu}}{x!}\), where \(e \approx 2.71818\) and \(\mu\) is the mean number of occurrences.