Todays Goals

1
Todays Goals

• Apply Bayes Rule
• Represent a discrete random variable as a formula
  or a histogram.
• Homework 4 (Due Wednesday Feb. 25) CH 3 5,12,
  18, Plus 2 questions on the web

2
Midterm

• Midterm in 2 weeks, Wed. March 4 in class
• One page of notes
• Probability reasoning questions like Web problem
  2 on HW 4 and the lightbulb problems
• A Bayes rule question
• Subjective probability and the axioms of
  probability
• Calculating probabilities using independence,
  multiplication rule, total probability, etc.

3
Bayes Theorem
4
Problem 5

• A part is supplied by three vendors. Of the
  total, 25 is supplied by Vendor 1, 25 by Vendor
  2 and 50 by Vendor 3. Their respective defect
  rates are 4, 2 and 1.
• Given that an inspected part is found defective,
  what is the probability that it came from Vendor
  1, 2 or 3?

5
Problem 5

• A part is supplied by three vendors. Of the
  total, 25 is supplied by Vendor 1, 25 by Vendor
  2 and 50 by Vendor 3. Their respective defect
  rates are 4, 2 and 1.
• Given that an inspected part is found defective,
  what is the probability that it came from Vendor
  1?
• P(1) .25 P(2) .25 P(3) .5
• P(D1) .04 P(D2) .02 P(D3) .01
• P(1D) ?

6
Practice Problem Glass manufacturing

• In a glass manufacturing process, expensive
  testing equipment can be used to detect the
  presence of extraneous gases in the furnace.
• These gases are the major cause of quality
  defects (bubbles) in the glass and are present in
  the furnace with probability 0.1.
• If these undesired gases are present, the
  equipment will give a warning signal with
  probability 0.99.
• If they are not present, the equipment may
  trigger a false alarm (i.e., it gives a warning
  signal even though no extraneous substances are
  present) with probability 0.05.
• Suppose the equipment is giving a warning signal,
  what is the probability that it is a false alarm
  (i.e. no extraneous gases are present)?

7
Practice Problem Glass manufacturing

• In a glass manufacturing process, expensive
  testing equipment can be used to detect the
  presence of extraneous gases in the furnace.
• These gases are the major cause of quality
  defects (bubbles) in the glass and are present in
  the furnace with probability 0.1.
• If these undesired gases are present, the
  equipment will give a warning signal with
  probability 0.99.
• If they are not present, the equipment may
  trigger a false alarm (i.e., it gives a warning
  signal even though no extraneous substances are
  present) with probability 0.05.
• Suppose the equipment is giving a warning signal,
  what is the probability that it is a false alarm
  (i.e. no extraneous gases are present)? Numeric
  answer, decimal

8
Practice Problem Glass manufacturing

• G is presence of gas
• W is a warning signal is given
• P(G) 0.1
• P(WG) 0.99
• P(WG) 0.05
• Suppose the equipment is giving a warning
  signal, what is the probability that it is a
  false alarm (i.e. no extraneous gases are
  present)?
• P(GW) P(WG)P(G)/P(WG)P(G)P(WG)P(G)
• 0.050.9 /0.050.9 0.99 0.1
  0.31!!

9
Random Variables

• The numerical outcome of a random experiment is
  called a random variable (r.v.)
• We denote random variables with upper case
  letter, X
• The observed numerical value once the experiment
  is run is denoted by the corresponding lower case
  letter, x
• A random variable is a rule associating a number
  with the elementary outcomes of an experiment.
• A random variable is a function whose domain is
  the sample space of the experiment, and range is
  the real numbers.

10
Random Variables

• The numerical outcome of a random experiment is
  called a random variable (r.v.)
• We denote random variables with upper case
  letter, X
• The observed numerical value once the experiment
  is run is denoted by the corresponding lower case
  letter, x
• Examples
• Random experiment Toss a coin 10 times
• Random variable X number of heads
• Random experiment Put an item on a life test
• Random variable 1 X lifetime of the item
• Random variable 2 X0 if lifetime gt10,000, X1
  otherwise

11
Observe that

• The random variable X associates a numerical
  value to each elementary outcome of the random
  experiment
• Many elementary outcomes may have the same
  numerical value associated.
• Example Sum of two dice
• The events corresponding to the distinct values
  of X are disjoint (mutually exclusive).
• The union of these events is the entire sample
  space (collectively exhaustive).
• A random variable defines a partition.

12
Types of Random Variables

• Discrete Random Variable
• takes a finite number of values or infinitely
  many values that can be arranged in a sequence
• Examples?
• Continuous Random Variable
• takes all values in an interval .
• Examples?

13
Random Variable

• Random Variables must be associated with a
  probability measure
• The rule for describing the probability measure
  associated with all values of a random variable
  is called a probability distribution.