Probability Distributions Background Discrete probability distributions Continuous probability distributions Multidimensional probability distributions

Probability Distributions Random (stochastic) variables Experimental measurements are not reproducible in a deterministic fashion Each measurement can be viewed a random variable X Defined on sample space S of an experiment Probability distribution Determines probability of particular events Discrete distributions: random variables are discrete quantities Continuous distributions: random variables are continuous quantities Cumulative probability distribution function F(x)

Discrete Probability Distributions Random variable X can only assume countably many discrete values: x1, x2, x3, … Probability density function f(x) Cumulative distribution function Properties

Discrete Distribution Example

Continuous Probability Distributions Random variable X can assume infinitely many real values Cumulative distribution function Probability density function Properties

Continuous Distribution Example Probability density function Cumulative distribution function Probability of events

Mean and Variance Discrete distribution Continuous distribution Symmetric distribution If f(c-x) = f(c+x), then f(x) is symmetric with respect to m = c Transformation of mean & variance Given random variable X with mean m & variance s2 The standardized random variable Z has zero mean & unity variance

Expectations and Moments Moments for continuous distributions Continuous distribution example

Binomial Distribution Governs randomness in games of chance, quality inspections, opinion polls, etc. X = {0,1,2,…,n} = number of times event A occurs in n independent trials Probability of obtaining A exactly x times in n trials Mean & variance: >> binopdf(x,n,p)  binomial probability >> binopdf(0,200,0.02)  ans = 0.0176 >> binocdf(x,n,p)  binomial cumulative probability >> 1 - binocdf(100,162,0.5)  ans = 0.0010

Poisson Distribution Infinitely many possible events Probability distribution function Limit of binomial distribution as Mean & variance: s2 = m Example Probability of a defective screw p = 0.01 Probability of more than 2 defects in a lot of 100 screws? Binomial distribution: m = np = (100)(0.01) = 1 Since p <<1, can use Poisson distribution to approximate solution Matlab functions: poisspdf(x,mu), poisscdf(x,mu)

Normal (Gaussian) Distribution Probability density function Cumulative distribution function Standardized normal distribution (m = 0, s2 = 1)

Computing Probabilities Interval probabilities Sigma limits Example X is a random variable with m = 0.8 & s2 = 4 Use Table A7 in text

Matlab: Normal Distribution Normal distribution: normpdf(x,mu,sigma) normpdf(8,10,2)  ans = 0.1210 normpdf(9,10,2)  ans = 0.1760 normpdf(8,10,4)  ans = 0.0880 Normal cumulative distribution: normcdf(x,mu,sigma) normcdf(8,10,2)  ans = 0.1587 normcdf(12,10,2)  ans = 0.8413 Inverse normal cumulative distribution: norminv(p,mu,sigma) norminv([0.025 0.975],10,2)  ans = 6.0801 13.9199 Random number from normal distribution: normrnd(mu,sigma,v) normrnd(10,2,[1 5])  ans = 9.1349 6.6688 10.2507 10.5754 7.7071

Multidimensional Probability Distributions Consider two random variable X and Y Two-dimensional cumulative distribution function Discrete distributions Continuous distribution Marginal distributions

Independent Random Variables Basic property Addition & multiplication of means Addition of variance Covariance