Distributions

The mean/expected value for the binomial distribution with parameters n,p is

np

The variance for the binomial distribution with parameters n,p {0≤p≤1} is

np(1-p)

The expected value for the gamma distribution with parameters s and 𝜆>0 is

s/𝜆

The variance for the gamma distribution with parameters s and 𝜆>0 is

s/𝜆²

The gamma function is defined by

Γ(a) = ∫_0^∞ e⁻ʸyᵃ⁻¹ dy

The normal distribution is symmetric about

μ

The mean/expected value for the Poisson distribution with parameter 𝜆>0 is

𝜆

The variance for the Poisson distribution with parameter 𝜆>0 is

𝜆

The variance for the exponential distribution with parameter 𝜆>0 is

𝜆⁻²

The expected value for the exponential distribution with parameter 𝜆>0 is

𝜆⁻¹

The expected value for the normal distribution with parameters 𝜇 and 𝜎² is

𝜇

The variance for the normal distribution with parameters 𝜇 and 𝜎² is

𝜎²

The variance for the geometric distribution with parameter p {0≤p≤1} is

(1-p)/(p²)

The expected value for the uniform distribution is

(a+b)/2

The variance for the uniform distribution is

(b-a)² / 12

Geometric Series

1/(1-p) = ∑_{k=0}^∞ p^k, for p in [0,1)

The mean/expected value for the geometric distribution with parameter p {0≤p≤1} is

1/p

Use a continuity correction when...

Approximating discrete distribution by a continuous distribution.

Integrate absolute value by

Breaking the integral up into pieces.

Bin(n,p) ~

Normal(p,√{n*p*(1-p)})

If X is normally distributed with parameters μ and 𝜎², then Y=aX+b is...

Normally distributed with parameters aμ+b and a²𝜎².

Mean is the same as

Expected value

Use the Poisson distribution by

Finding the expected value.

DeMoivre-Laplace limit theorem

If S_n denotes the number of successes that occur when n independent trials, each resulting in a success with probability p, are performed, then, for any a < b, P{a ≤ (S_n - np)/√{n*p*(1-p)}) ≤ b} = 𝜙(b) - 𝜙(a) As n -> ∞

Given a pdf with two variables and the expected value of the variable, e.g. f_X(x) = ax+b for x in (0,1) and E(X) = 6, you can determine a AND b by

Solving the system of linear equations: { ∫_R (f(t))dt = 1, ∫_R t*(f(t))dt = 6 }

Solving minimization problems

Take the derivative and solve = 0

If X is normally distributed with parameters μ and σ², then

Z = (X − μ)/σ is normally distributed with parameters 0 and 1.

Variance for a discrete random variable X is given by Variance for a continuous random variable Y is given by

Var(X) = E(X²) - E²(X) and Var(Y) = E(Y²) - E²(Y)

A random variable X has a Poisson distribution if...

X = number of events occurring during a time interval of length 1, given that events occur at an average rate of λ per unit time. NEperT

A random variable X has a binomial distribution if...

X = number of successes in K independent Bernoulli trials. NSinK

A random variable X has a negative binomial distribution if...

X = number of trials for achieving r successes. NT4KS

A random variable X has a geometric distribution if

X = number of trials for the first success, in an infinite sequence of Bernoulli trials. NT41S

A random variable X has a Normal distribution if...

X = random fluctuation arising from many causes.

A random variable X has a Uniform distribution if...

X = randomly chosen point in the interval (a, b).

A random variable X has a Bernoulli distribution if...

X = result of a single Bernoulli trial.

A random variable X has an Exponential distribution if...

X = waiting time until the first event, if there are λ events per unit time on average. WT41E

A random variable X has a Gamma distribution if...

X = waiting time until the α'th event (when α is a positive integer), if there are λ events per unit time on average. WT4aE

If X is normal with parameters μ = 0 and 𝜎²=1, then

X is called standard. i.e. X is a standard normal distribution.

Exponential Taylor series

e^x = ∑_{k=0}^∞ (n^k)/k!

The probability density function for the normal distribution with parameters 𝜇 and 𝜎² is

e^{-(x-𝜇)²/2𝜎²}/(sqrt(2π)𝜎) DEFINED EVERYWHERE

The probability density function for the uniform distribution over the open interval (a,b) is

f(x) = 1/(b-a) if a<x<b and f(x) = 0 otherwise.

The probability density function for the gamma distribution with parameters s and 𝜆>0 is

f(x) = 𝜆e^{-𝜆x} (𝜆x)ˢ⁻¹ Γ(s)⁻¹ if x≥0 and f(x) = 0 if x < 0

The probability density function for the exponential distribution with parameter 𝜆>0 is

f(x) = 𝜆e^{-𝜆x} if x≥0 and f(x) = 0 if x<0

The probability mass function for the Poisson distribution with parameter 𝜆>0 is

f_X(t) = (e^{-𝜆})𝜆ᵗ (t!)⁻¹ t is any whole number

The probability mass function for the binomial distribution with parameters n,p is

f_X(t) = nCr(n, t)pᵗ(1-p)ⁿ⁻ᵗ t is any whole number

The probability mass function for the geometric distribution with parameter p {0≤p≤1} is

f_X(t) = p(1-p)ᵗ⁻¹, t is a natural number i.e. probability of success times probability of t-1 failures.