Download MATH408: PROBABILITY & STATISTICS

MATH408: Probability & Statistics
Summer 1999
WEEK 5
Dr. Srinivas R. Chakravarthy
Professor of Mathematics and Statistics
Kettering University
(GMI Engineering & Management Institute)
Flint, MI 48504-4898
Phone: 810.762.7906
Email: [email protected]
Homepage: 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   f ( x, y) dx dy 1
P(a  X  b, c  Y  d )  
b
a

d
c
f ( x, y) dx dy
Measures of Joint PDF
Independence
We say that two random variables X and Y are
independent if and only if
P(XA, YB) = P(XA)P(YB) 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 function
function 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?
ILLUSTRATIVE EXAMPLE
UTT: Ho: 0.1 vs H1:  > 0.1
LTT: Ho: 0.1 vs H1:  < 0.1
UTT
LTT
P(Type 1 error)
To conclude > 0.1
when in fact 0.1.
To conclude < 0.1
when in fact 0.1.
P(Type 2 error)
To conclude 0.1
when in fact > 0.1.
To conclude 0.1
when in fact < 0.1.
Test on µ using normal
• Sample size is large
• Sample size is small, population is
approximately normal with known .
TTT: Ho: = 0 vs H1: 0
DNR Region
CP_1
µ
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 ˆ denote its
UMV estimator.
• Given , 100(1-  )% CI for
 is
constructed using the sampling (probability)
distribution of ˆ as follows.
• Find L and U such that P(L < ˆ< U) = 1.
ˆ
• Note that L and U are functions of .