Discrete and Continuous Distributions G. V. Narayanan

Discrete Probability Distributions 1.Bernoulli Probability Function 2.Binomial Probability Function 3.HyperGeometric Probability Function 4.Poisson Probability Function

Continuous Probability Distributions 1.Uniform Probability Function 2.Normal Probability Function 3.Standard Normal Probability Function 4.LogNormal Probability Function 5.Exponential Probability Function 6.Geometric Probability Function 7.Weibull Probability Function

Understand Random Variable A Random variable is a NUMERIC VALUE assigned to a ‘quantity’ or ‘property to an Object or item’ Examples of QUANTITY: Quantity can be an ‘item’ or ‘property of any object’ or ‘length’ or ‘width’ or ‘thickness’ or ‘Area’ or ‘Answers to a Questionaire’ or ‘any scientific numeric values’ Random Variable ‘Value’ can be either ‘DISCRETE’ or ‘Continuous Interval’ Mostly, a Random Variable is represented by symbol ‘X’ (Upper case Letter, never Lower case letter) Mostly, a Random Variable ‘Value’ is represented by symbol ‘x’ (Lower case letter, never Upper case letter)

Random Variable Values of a distribution Examples of Random Variable Values: ‘Discrete’ Random Variable Values for the toss of a COIN: Head or Tail => Assigned Values are ‘0’ (zero) for Tail (or Head) or ‘1’ (one) for Head (or Tail) ‘Discrete’ Random Variable Values for the roll of a Dice: Face up 1, Face up 2, Face up 3, Face up 4, Face up 5, Face up 6 => Assigned Values are ‘1’ (one) for Face up 1 etc Discrete Random Variable Values for the number of New cars sold is any positive number, 0,1,2,3, … Discrete Random Variable Values for the number of students is an positive number less than Max Value

Random Variable Values of a distribution Examples of Random Variable Values: ‘Continuous’ Random Variable Values for the height of students in a university: Any Positive Real valued number in an interval, say between (3 feet and 7 feet) with a decimal or in feet and inches ‘Continuous’ Random Variable Values for the impurities in a liquid in units of parts per millions: any Positive real values number in an interval, say between 3 and 10 PPM ‘Continuous’ Random Variable Values for the Length or Diameter of Rods: any positive real values between 0 and maximum value

About Population Parameters Each Probability Distribution has either ONE or TWO or Three population parameters

The Population Parameters of a Distribution We always talk about Either ‘Population’ Or ‘Sample’ Data from measurements or from ‘Population’ Data We will ALWAYS discuss: 1) ‘Probability’ ‘Mass’ or ‘Density’ ‘Distribution’ Functions; 2) ‘Cumulative’ Probability Distributions; 3) ‘Inverse’ Cumulative Probability Distributions For a GIVEN set of ‘Population’ Parameters These population parameters are NOT the SAME for ALL Distributions

The Population Parameters of a Distribution For Bernoulli Distribution, the probability of ‘success’ or ‘failure’ or ‘defective’ etc is the ONLY population parameter, denoted by symbol ‘p’ For Binomial Distribution, the TWO population parameters are: N (Total data count) and ‘p’ of Random Variable at a value For Poisson Distribution, the only population parameter is denoted by symbol ‘lamda’, this lamda equals ‘N*p’ for approximating Binomial distribution for N > 20 and p < 0.05 Note: Text treats Poisson as Discrete Distribution where as Poisson can be used for Continuous Random Variable in an interval

The Population Parameters of a Distribution HyperGeometric Distribution has THREE parameter: N( Total Data count), n (Sample Data count) and k (number of successes within ‘n’ selected samples of N items) The Normal Distribution has TWO population parameters: Mean value ‘mu’ and std deviation ‘sigma’. For Standard Normal Distribution, Mean = 0 and Std dev = 1

The Population Parameters of a Distribution LogNormal Distribution has Two Parameters: Mean and Std Dev Exponential Distribution has One Parameter Weibull Distribution has Two Parameters, Alpha and Beta

Discrete RV Probability Distributions 1.Binomial Distribution 2.Hypergeometric Distribution 3.Poisson Distribution 4.Geometric Distribution

Bernoulli Distribution Function  Bernoulli Trials are independent (assumed) p(success) = p p(Failure)=1-p  Random Variable X ~ Bernoulli(p)  Probability Mass Function of X is: p(1) = P(X=1) = p p(0) = P(X=0) = 1-p [Random Variable Values are Discrete Values ‘0’ and ‘1’]  Mean = p  Variance = p*(1-p)

Binomial Distribution Function

Geometric Distribution Function

Hypergeometric Distribution Function

Poisson Distribution Function

Discrete Probability Values Computing Examples

Computing Probability of Binomial Distribution

Computing Probability of Hypergeometric Distribution

Computing Probability of Poisson Distribution

Continuous RV Probability Distributions 1.Uniform Distribution 2.Standard Normal Distribution 3.Normal Distribution 4.LogNormal Distribution 5.Exponential Distribution 6.Weibull Distribution

Uniform Distribution Function

Standard Normal Distribution Function

Normal Distribution Function

LogNormal Distribution Function

Exponential Distribution Function

Weibull Distribution Function

Continuous Probability Values Computing Examples

Computing Probability of Uniform Distribution

Computing Probability of Standard Normal Distribution

Computing Probability of Normal Distribution

Computing Probability of LogNormal Distribution

Computing Probability of Exponential Distribution

Compute Uncertainity of Probability Distribution Mean and Variance If Population parameters are unknown, compute uncetainity on parameters computed using SAMPLE data.

Sample Data Values of Population Parameters If Population parameters are UNKNOWN, then Sample Data is used to compute Equivalent Population Parameters, For Example, if Mean of Population is UNKNOWN, the mean of Sample (s) can be used as equivalent to Mean of Population (Mu) Same reasoning goes for Standard Deviation value Std Dev of Sample Data can be used as Std Dev value of a population. The UNCERTANITY of error due to using Sample for obtaining Population parameters must be COMPUTED

Computing Uncertainty of Mean and Standard Deviation for a Binomial Distribution

Check on Data to find its Distribution

Normal Probability Plot Read Section 4.10

Central Limit Theorem

Simulation of Data for a Given Distribution

MiniTab Use in Computing Probability for Binomial Distribution Use Menu Calc  Probability Distributions  Binomial

MiniTab Use in Computing Probability for Poisson Distribution Use Menu Calc  Probability Distributions  Poisson

MiniTab Use in Computing Probability for Hypergeometric Distribution Use Menu Calc  ProbabilityDistributions  Hypergeomet ric See text Page 232

MiniTab Use in Computing Probability for Standard Normal Distribution Use Menu Calc  Probability Distributions  Normal

Minitab use to compute Inverse Cumulative Probability for Standard Normal Distribution Distribution Calc  Probability Distributions  Normal