Known Probability Distributions

1
Known Probability Distributions

• Engineers frequently work with data that can be
  modeled as one of several known probability
  distributions.
• Being able to model the data allows us to
• model real systems
• design
• predict results
• Key discrete probability distributions include
• binomial
• negative binomial
• hypergeometric
• Poisson

2
Binomial Multinomial Distributions

• Bernoulli Trials
• Inspect tires coming off the production line.
  Classify each as defective or not defective.
  Define success as defective. If historical data
  shows that 95 of all tires are defect-free, then
  P(success) 0.05.
• Signals picked up at a communications site are
  either incoming speech signals or noise. Define
  success as the presence of speech. P(success)
  P(speech)
• Bernoulli Process
• n repeated trials
• the outcome may be classified as success or
  failure
• the probability of success (p) is constant from
  trial to trial
• repeated trials are independent

3
Binomial Distribution

• Example
• Historical data indicates that 10 of all bits
  transmitted through a digital transmission
  channel are received in error. Let X the number
  of bits in error in the next 4 bits transmitted.
  Assume that the transmission trials are
  independent. What is the probability that
• Exactly 2 of the bits are in error?
• At most 2 of the 4 bits are in error?
• More than 2 of the 4 bits are in error?
• The number of successes, X, in n Bernoulli trials
  is called a binomial random variable.

4
Binomial Distribution

• The probability distribution is called the
  binomial distribution.
• b(x n, p) , x 0, 1, 2, , n
• where p probability of success
• q probability of failure 1-p
• For our example,
• b(x n, p)

5
For Our Example

• What is the probability that exactly 2 of the
  bits are in error?
• At most 2 of the 4 bits are in error?
• More than 2 of the 4 bits are in error?

6
Expectations of the Binomial Distribution

• The mean and variance of the binomial
  distribution are given by
• µ np
• s2 npq
• Suppose, in our example, we check the next 20
  bits. What are the expected number of bits in
  error? What is the standard deviation?
• µ 20 (0.1) 2
• s 2 20 (0.1) (0.9) 1.8 s 1.34

7
Another example

• A worn machine tool produces 1 defective parts.
  If we assume that parts produced are independent,
  what is the mean number of defective parts that
  would be expected if we inspect 25 parts?
• µ 25 (0.01) 0.25
• What is the expected variance of the 25 parts?
• s 2 25 (0.01) (0.99) 0.2475
• Note that 0.2475 does not equal 0.25.

8
Helpful Hints

• Suppose we inspect the next 5 parts b(x 5,
  0.01)
• Sometimes it helps to draw a picture.
• P(at least 3) ? ________________
• 0 1 2 3 4 5
• P(2 X 4) ? ________________
• 0 1 2 3 4 5
• P(less than 4) ? ________________
• 0 1 2 3 4 5
• Appendix Table A.1 (pp. 726-731) lists Binomial
  Probability Sums, ? rx0 b(x n, p)