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

2. 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)

3. 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

4. Discrete Distribution Example

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

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

7. 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

8. Expectations and Moments
Moments for continuous distributions
Continuous distribution example

9. 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 =
>> binocdf(x,n,p)  binomial cumulative probability
>> 1 - binocdf(100,162,0.5)  ans =

10. 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)

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

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

13. Matlab: Normal Distribution
Normal distribution: normpdf(x,mu,sigma)
normpdf(8,10,2)  ans =
normpdf(9,10,2)  ans =
normpdf(8,10,4)  ans =
Normal cumulative distribution: normcdf(x,mu,sigma)
normcdf(8,10,2)  ans =
normcdf(12,10,2)  ans =
Inverse normal cumulative distribution: norminv(p,mu,sigma)
norminv([ ],10,2)  ans =
Random number from normal distribution: normrnd(mu,sigma,v)
normrnd(10,2,[1 5])  ans =

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

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