Set Equations

`P(A|B) = (P(A nn B)) / (P(B))` provided `P(B)!=0`

` = (|A nn B|)/(|B|)`

`"Bayes Theorem" = P(B|A) = (P(A|B)*P(B))/(P(A))`

`(A uu B)^c = A^c nn B^c`

`(A nn B)^c = A^c uu B^c`

If A is independent of B:

`P(A nn B) = P(A)*P(B)`

`P(A|B) = P(A)`

`P(B|A) = P(B)`

`"Addition of probabilities" = P(A) = P(A|B) * P(B) + P(A|B^c)*P(B^c)`

PA|B=P(AB)P(B)   provided P(B)0             = |AB||B|Bayes Theorem = P(B|A) = P(A|B)·P(B)P(A)(AB)c=AcBc(AB)c=AcBcA independant of B:    P(AB)=P(A)·P(B)    P(A|B)=P(A)    P(B|A)=P(B)Addition of probabilities:    P(A) = P(A|B)·P(B) + P(A|Bc)·P(Bc) mi/mi/mi/mi/mo/mimi

Bernoulli Distribution

Range X : 0, 1

P(X=1) = p     P(X=0) = 1-pX ~ Bernoulli(p) or Ber(p)Ber(p) = Bin(1,p)E(X) = pVar(X)=(1-p)p

Binomial Distribution

This is the sum of independent Bernoulli(p) variables. Range X : 0, 1, … n

X ~ Binomial(n,p)p(k)=Bin(n,k)=nkpk(1-p)n-kBin(n,n) = pnE(X) = np

Var(X)=np(1-p)

Geometric Distribution

Range X : 0, 1, 2, … Var(X) = (1-p) / p^2

X ~ geometric(p) or geo(p)p(k) = P(X=k) = (1-p)kpE(X) = 1-ppP(X=n+k|Xn) = P(X=k)

Uniform Distribution

Where all outcomes are equally likely. Range X : 1, 2, ..., n p(k) = 1/n E(X) = (n+1) / 2 Var(X) = (n^2-1)/12

X ~ uniform(N)

Exponential Distribution

Models: Waiting times

X ~ exponential(`lambda`) or exp(`lambda`)

Parameter: `lambda` (called the rate parameter)

  • Range: `[0,oo)`
  • Density: `f(x) = lambda e^(-lambda x)` for `0 <= x`
  • `F(x) = 1 - e^(-lambda x)`

Normal distribution

  • Parameters: `mu` `sigma`
  • Range: `(-oo,oo)`
  • Notation: `"normal"(mu,sigma^2)` or `N(mu,sigma^2)`
  • Density: `f(x)=1/(sigma sqrt(2pi)) e^(-(x-mu)^2/(2sigma^2))`
  • Distribution: F(x) has no formula, so use tables or software such as pnorm in R to compute F(x).
    • pnorm(.6,0,1) returns the .6 quantile of the standard normal distribution.
  • Models: Measurement error, intelligence/ability, height, averages of lots of data.
  • Standard Normal Cumulative Distribution : N(0,1) = `Phi(z)` : has mean 0 and variance 1.
  • Standard Normal Density: `phi(z) = 1/sqrt(2pi) e^(-x^2/2)`
  • `N(mu,sigma^2)` has mean `mu`, variance `sigma^2`, and standard deviation `sigma`.
  • `P(-1 <= Z <= 1) ~~ .68`, `P(-2 <= Z <= 2) ~~ .95`, `P(-3 <= Z <= 3) ~~ .99`
  • `P(Z <= 1) ~~ .84`, `P(Z <= 2) ~~ .975`, `P(Z <= 3) ~~ .995`
  • `Phi(x) = P(Z <= x)`

Discrete Random Variables

Random variable X assigns a number to each outcome: `X : Omega -> R`

`X = a " denotes event " {omega | X(omega) = a}`

`"probability mass function (pmf) of X is given by: " p(a) = P(X=a)`

`"Cumulative distribution function (cdf) of X is given by: " F(a) = P(X<=a)`

Continuous random variables

`"Probability density function (pdf)" = P(c<=x<=d) = int_c^d f(x) dx "for" f(x)>=0`

`"Cumulative distribution function (cdf)" = F(x) = P(X<=x) = int_-oo^x f(t) dt`

Properties of the cdf (Same as for discrete distributions)

  • (Definition) `F(x) = P(X<=x)`
  • `0 <= F(x) <= 1`
  • non-decreasing
  • `lim_(x->-oo) F(x) = 0`
  • `lim_(x->oo) F(x) = 1`
  • `P(c < X <= d) = F(d) - F(c)`
  • `F'(x) = f(x)`

Expected Value (mean or average)

  • weighted average = `E(X) = sum_(i=1)^n x_i * p(x_i)`
  • `E(X+Y) = E(X) + E(Y)`
  • `E(aX+b) = a*E(X) + b`
  • `E(h(X)) = sum_i h(x_i) * p(x_i)`
  • `E(X) = int_a^b x * p(x) dx` (units for p(x) are probability/dx).

Variance

  • `"mean" = E(X) = mu`
  • variance of X = `Var(X) = E((x-mu)^2) = sigma^2 = sum_(i=1)^n p(x_i)(x_i-mu)^2`
  • standard deviation = `sigma = sqrt(Var(X))`
  • `Var(aX+b) = a^2 Var(X)`
  • `Var(X) = E(X^2) - E(X)^2`
  • If X and Y are independent then: `Var(X+Y) = Var(X) + Var(Y)`

Quantiles

  • median is x for which `P(X<=x) = P(X>=x)`
  • median is when cdf `F(x) = P(X<=x) = .5`
  • The pth quantile of X is the value `q_p` such that `F(q_p)=P(X<=q_p)=p`. In this notation `q_.5` is the median.

Central Limit Theorem & Law of Large Numbers

  • LoLN: As n grows, the probability that E(X) is close to `mu` goes to 1.
  • LoLN: `lim_(n->oo) P(|E(X) - mu| < alpha) = 1`
  • CLT: As n grows, the distribution of E(X) converges to the normal distribution `N(mu,sigma^2/n)`.
  • Z = standardization of X = `(X - mu)/sigma`
  • Z has mean 0, standard deviation 1
  • If X has a normal distribution, Z is the standard normal distribution
Version 58.1 last modified by Geoff Fortytwo on 13/05/2015 at 11:25

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