MATH408: Probability & StatisticsSummer 1999

WEEK 5

Dr. Srinivas R. ChakravarthyProfessor of Mathematics and Statistics

Kettering University(GMI Engineering & Management Institute)

Flint, MI 48504-4898Phone: 810.762.7906

Email: schakrav@kettering.eduHomepage: www.kettering.edu/~schakrav

Joint PDF

• So far we saw one random variable at a time. However, in practice, we often see situations where more than one variable at a time need to be studied.

• For example, tensile strength (X) and diameter(Y) of a beam are of interest.

• Diameter (X) and thickness(Y) of an injection-molded disk are of interest.

Joint PDF (Cont’d)X and Y are continuous

• f(x,y) dx dy = P( x < X < x+dx, y < Y < y+dy) is the probability that the random variables X will take values in (x, x+dx) and Y will take values in (y,y+dy).

• f(x,y) 0 for all x and y and

1),( dydxyxf

d

c

b

adydxyxfdYcbXaP ),(),(

Measures of Joint PDF

Independence

We say that two random variables X and Y are independent if and only if

P(XA, YB) = P(XA)P(YB) for all A and B.

EXAMPLES

Groundwork for Inferential Statistics

• Recall that, our primary concern is to make inference about the population under study.

• Since we cannot study the entire population we rely on a subset of the population, called sample, to make inference.

• We saw how to take samples.

• Having taken the sample, how do we make inference on the population?

Basic Concepts

Figure 3-36 (a) Probability density function of a pull-off force

measurement in Example 3-33.

Figure 3-36 (b) Probability density function of the average of 8 pull-off force measurements in Example 3-33.

Figure 3-36 (c) Probability density Probability density functionfunction of the sample variance of 8 pull-off force

measurements in Example 3-33.

An important result

Examples

Central Limit Theorem

• One of the most celebrated results in Probability and Statistics

• History of CLT is fascinating and should read “The Life and Times of the Central Limit Theorem” by William J. Adams

• Has found applications in many areas of science and engineering.

CLT (cont’d)

• A great many random phenomena that arise in physical situations result from the combined actions of many individual ones.

• Shot noise from electrons; holes in a vacuum tube or transistor; atmospheric noise, turbulence in a medium, thermal agitation of electrons in a conductor, ocean waves, fluctuations in stock market, etc.

CLT (cont’d)

• Historically, the CLT was born out of the investigations of the theory of errors involved in measurements, mainly in astronomy.

• Abraham de Moivre (1667-1754) obtained the first version.

• Gauss, in the context of fitting curves, developed the method of Least Squares, which lead to normal distribution.

Examples

HOMEWORK PROBLEMS

Sections 3.11 through 3.12

109,111, 114-116-119, 121-123, 129-130

Examples

Tests of Hypotheses

•Two types of hypotheses: Null (H0)and alternative (H1)

Basic Ideas in Tests of Hypotheses

• Set up H0 and H1. For a one-sided case, make sure these are set correctly. Usually these are done such that type 1 error becomes “costly” error.

• Choose appropriate test statistic. This is usually based on the UMV estimator of the parameter under study.

• Set up the decision rule if = P(type 1 error) is specified. If not, report a p-value.

• Choose a random sample and make the decision.

Setting up Ho and H1

• Suppose that the manufacturer of airbags for automobiles claims that the mean time to inflate airbag is no more than 0.1 second.

• Suppose that the “costly error” is to conclude erroneously that the mean time is < 0.1.

• How do we set up the hypotheses?

P(Type 1 error) P(Type 2 error)UTT To conclude > 0.1

when in fact 0.1.To conclude 0.1when in fact > 0.1.

LTT To conclude < 0.1when in fact 0.1.

To conclude 0.1when in fact < 0.1.

UTT: Ho: 0.1 vs H1: > 0.1LTT: Ho: 0.1 vs H1: < 0.1

ILLUSTRATIVE EXAMPLE

Test on µ using normal

• Sample size is large

• Sample size is small, population is approximately normal with known .

DNR Region

TTT: Ho: = 0 vs H1: 0

µCP_1 CP_2

Computation of P(type 2 error)

Example (page 142)

•µ = Mean propellant burning rate (in cm/s).

•H0:µ = 50 vs H1:µ 50.

•Two-sided hypotheses.

•A sample of n=10 observations is used to test the hypotheses.

•Suppose that we are given the decision rule.

•Question 1: Compute P(type 1 error)

•Question 2: Compute P(type 2 error when µ =52.

DECISION RULE

Calculation of P(type 1 error)

Example

Confidence Interval

• Recall point estimate for the parameter under study.

• For example, suppose that µ= mean tensile strength of a piece of wire.

• If a random sample of size 36 yielded a mean of 242.4psi.

• Can we attach any confidence to this value?• Answer: No! What do we do?

Confidence Interval (cont’d)

• Given a parameter, say, , let denote its UMV estimator.

• Given , 100(1- )% CI for is constructed using the sampling (probability) distribution of as follows.

• Find L and U such that P(L < < U) = 1- .• Note that L and U are functions of .