Probability Distributions

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(cx), 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
• gtgt binopdf(x,n,p) ? binomial probability
• gtgt binopdf(0,200,0.02) ? ans 0.0176
• gtgt binocdf(x,n,p) ? binomial cumulative
  probability
• gtgt 1 - binocdf(100,162,0.5) ? ans 0.0010

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 ltlt1, 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 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

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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