1. Covariance/ Correlation
A measure of the nature of the association between two variables
Describes a potential linear relationship
Positive relationship
Large values of X result in large values of Y
Negative relationship
Large values of X result in small values of Y
“Manual” calculations are based on the joint probability distributions
See examples in Chapter 4 (pp )
MDH Chapter 4 - 5
EGR

2. 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
Note that the 9th edition eliminated section 5.2 Discrete Uniform Distribution
Section 5.2 is now binomial and multinomial
MDH Chapter 4 - 5
EGR

3. 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
MDH Chapter 4 - 5
EGR

4. 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.
remember, YOU define what a “success” is … e.g., votes, defects, errors
MDH Chapter 4 - 5
EGR

5. 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) =
p =
q =
b(x; n, p) = (4 choose x)(0.1)x(0.9)4-x , x = 0,1,2,3,4
MDH Chapter 4 - 5
EGR

6. 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?
b(x; n, p) = (4 choose x)(0.1)x(0.9)4-x , x = 0,1,2,3,4
P(X = 2) = (4 choose 2) (0.1)2(0.9)2 =
P(X < 2) = P(0) + P(1) + P(2) = (4 choose 0) (0.1)0(0.9)4 + (4 choose 1) (0.1)1(0.9)3 +
(4 choose 2) (0.1)2(0.9)2 =
MDH Chapter 4 - 5
EGR

7. Expectations of the Binomial Distribution
The mean and variance of the binomial distribution are given by
μ = np
σ2 = 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
σ 2 = 20 (0.1) (0.9) = σ = 1.34
μ = np = 20(0.1) = 2
σ2 = npq = 20(0.1)(0.9)= 1.8, σ = 1.34
MDH Chapter 4 - 5
EGR

8. 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?
σ 2 = 25 (0.01) (0.99) =
Note that does not equal 0.25.
μ = np = 25(0.01) = 0.25
σ2 = npq = 25(0.01)(0.99)= **
NOTE: ≠ 0.25
MDH Chapter 4 - 5
EGR

9. 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)  ________________
P(2 ≤ X ≤ 4)  ________________
P(less than 4)  ________________
Appendix Table A.1 (pp ) lists Binomial Probability Sums, ∑ rx=0 b(x; n, p)
MDH Chapter 4 - 5
EGR