Notes for Probability and Statistics

Sample mean: is a numerical average.

\overline{x} = \sum_{i = 1}^{n} \frac{x_{i}}{n}

Sample median: is the midpoint in an ordered set of numbers.

For x_{1}, x_{2}, x_{3}, ..., x_{n} where x_{i} \leq x_{i + 1}

\tilde{x} = {\begin{matrix} x_{\frac{n + 1}{2}} & ; n = k c + 1 \\ \frac{x_{\frac{n}{2}} + x_{\frac{n}{2} + 1}}{2} & ; n = k c \end{matrix} k \in ℕ

Permutations: is the number of ways to arrange a set of objects.

The number of permutations of n distinct objects taken r at a time is = \frac{n!}{(n - r)!}

Combinations: is the number of ways to select objects.

The number of combinations of n object taken r at a time is (\binom{n}{r}) = \frac{n!}{r! (n - r)!}

Chebyshev's theorem:

P (μ - k σ < X < μ + k σ) \leq 1 - \frac{1}{k^{2}}

Bernoulli process:

The experiment consists of n repeated trials
Each trial results in an outcome that may be classified a success or a failure
The probability of a success, denoted by p, remains constant from trial to trial
The repeated trials are independant

Binomial distribution: is the distribution of a binomial random variable which is the number of success in n Bernoulli trials.

b (x; n, p) = (\binom{n}{x}) p^{x} q^{n - x}; x = 0, 1, 2, ..., n

μ = n p

σ^{2} = n p q

Poisson process:

The number of outcomes occurring in one time interval or specified region is independant of the number that occurs in any other disjoint time interval or region of space
The outcome that a single outcome will occur during a very short time interval or in a small region is proportional to the length of the time interval or size of the region and does not depend on the number of outcomes occurring outside this time interval or region
The probability that more than one outcome will occur in such a short time interval or fall in such a small region is negligible

Poisson distribution: is the distribution of a Poisson random variable which is the number of outcomes occuring in a Poisson experiment.

p (x; λ t) = \frac{e^{- λ t} {(λ t)}^{x}}{x!}, x = 0, 1, 2, ...

μ = σ^{2} = λ t

Normal distribution: is also called a Gaussian distribution and it is the distribution of a continuous random variable with a bell shaped distribution.

n (x; μ, σ) = \frac{1}{\sqrt{2 π} σ} e^{- \frac{1}{2} {(\frac{x - μ}{σ})}^{2}}

Gamma function:

Γ (α) = \int_{0}^{∞} x^{α - 1} e^{- x} dx = (α - 1) Γ (α - 1)

Γ (n) = (n - 1)!; n \in ℕ

Γ (\frac{1}{2}) = \sqrt{π}

Gamma distribution: in the Poisson process, λ is the mean number of events per interval. In the gamma distribution, λ = 1 / β. β is the mean time between events. α is the number of events to occur.

For instance in the problem:

Suppose the telephone calls arriving at a switchboard come at 5 per minute. What is the likelihood that up to a minute will elapse until 2 calls come in?

λ = 5, β = 1 / 5, α = 2

f (x) = {\frac{1}{β^{α} Γ (α)} x^{α - 1} e^{\frac{- x}{β}}; x > 0, α > 0, β > 0

μ = α β

σ^{2} = α β^{2}

Expodential distribution: is a gamma distribution with α = 1. This will give the probability of failure for a given time.

Chi-square distribution: is a gamma distribution with β = 2 and α = v / 2 where v ∈ ℕ called the degrees of freedom.

Lognormal distribution: applies where when a natural log transformation of the random variable results in a normal distribution.