Useful Probability Distributions

Transcription

1 Useful Probability Distributions Standard Normal Distribution Binomial Multinomial Hypergeometric Poisson Beta Binomial Student s t Beta Gamma Dirichlet Multivariate Normal and Correlation

2 Standard Normal Distribution X 2 ~ N, σ Standardization Z X / σ A general normally distributed random variable is transformed into one which has a standard normal distribution EZ0 and VarZ Division by σ ensures the resulting statistic is dimensionless PrZ<z is denoted Φz: cumulative distribution function Tabulated, e.g., z.6449, Φz0.950, z2.5758, Φz0.995 Φ-z- Φz

3 Student s t-distribution Similar to Standard Normal Distribution Standard deviation σ of a Normal Distribution is rarely nown T-statistic taes into account uncertainty associated with estimating σ If X i 2 has a normal distribution with X ~ N, σ / n If X i 2 ~ N, σ, n S is an estimate of then X / S X which has a i / std dev n has a t - distribution greater dispersion than standard normal

4 Binomial Models a sequence of independent trials in which there are only two possible mutually eclusive outcomes n is the number of trials X is the number of successes p is the probability of success in any individual trial, and q-p PX, 0,..,n is denoted p i Pr X n p n p n Distribution of X is denoted in shorthand as X ~ Binn,p If X is number of sies in 0 throws of a fair die, then X~Bin0,/6 EXnp, VarXnpq q n p

5 Normal Approimation to the Binomial Binomial is a discrete distribution where it is tedious to evaluate eact probabilities for large number of events, e.g., probability of 530 or fewer heads in 000 tosses of a fair coin X~Nnp,npq Pr X Pr Z 530 n 000, 2 If X ~ N µ, σ Z X µ / σ is N0, p 0.5 < z is denoted Φ z Φ n / 250 Φ Z has a standard normal distribution which is tabulated

6 Multinomial Distribution Generalization of Binomial Models a sequence of independent trials where there are possible mutually eclusive outcomes...!!..!!,.., Pr 2 i i p p p n X X

7 Hypergeometric Distribution Binomial is with replacement Models a sequence of independent trials where there are 2 possible mutually eclusive outcomes Hypergeometric is without replacement E.g., Probability of the number X of illicit tablets in a sample of size m from a consignment of size N in which R are illicit and N-R are licit is PrX If N20, R0, m6 then PrX3 0 C 3 0 C 3 / 20 C R N R m N m

8 Beta-Binomial Distribution Consignment of tablets, a proportion of which are suspected drugs. For large consignments, probability distribution of the proportion t which are drugs can be modeled with a beta distribution, which treats the proportion t as a variable which is continuous over the interval 0, For small consignments, say N<50, a more accurate distribution, which recognizes the discrete nature of possible values of the proportions is used

9 Beta Distribution Consignment of N tablets, No of illicit is R Proportion of illicit is R/N which has a finite no of values ranging from 0/N to N/N in steps of /N As N increases proportion becomes closer to a continuous measurement over interval 0, Modeled by a beta distribution Denote true proportion by random variable 0 < < p /, β where B, β B, β β Γ Γ β nown as the Γ + β Γ is the gamma function defined Γ +! Γ / 2 π beta as function Values of and β reflect prior beliefs before inspection Bayesian philosophy Large value of relative to β would imply a belief that was high Neutral belief would have β

10 Beta Distribution It is a general type of statistical distribution which is related to the gamma distribution. Beta distributions have two free parameters, which are labeled and The domain is [0,], and the probability function P and distribution function D are given by where Ba,b is the beta function, is the regularized beta function, and

11 Gamma Distribution Probability Density function epressed in terms of te Gamma function Alternatively epressed as 0, ; / > Γ for e f 0, ; > Γ for e g β β β

12 Modeling distributions of distances in writer verification Gamma Gaussian

13 Dirichlet Distribution Generalization of Beta distribution to categories analogous to generalization of binomial distribution to multinomial distribution Eample: proportion of illicit drugs when there are types of drugs Given a consignment of size N No of tablets of each type is R i, i.., Proportions are R i /N As N increases the proportions are continuous over 0,

14 Dirichlet Distribution Characterized by parameters {,.. } chosen to represent prior beliefs about proportions....,.., 0,....,.. i i i B where B f + Γ Γ Γ < <