Introduction%20to%20Probability%20and%20Statistics

1
Introduction to Probability and Statistics
Chapter 12
2
Topics

• Types of Probability
• Fundamentals of Probability
• Statistical Independence and Dependence
• Expected Value
• The Normal Distribution

3
Sample Space and Event

• Probability is associated with performing an
  experiment whose outcomes occur randomly
• Sample space contains all the outcomes of an
  experiment
• An event is a subset of sample space
• Probability of an event is always greater than or
  equal to zero
• Probabilities of all the events must sum to one
• Events in an experiment are mutually exclusive if
  only one can occur at a time

4
Objective Probability

• Objective Probability
• Stated prior to the occurrence of the event
• Based on the logic of the process producing the
  outcomes
• Relative frequency is the more widely used
  definition of objective probability.
• Subjective Probability
• Based on personal belief, experience, or
  knowledge of a situation.
• Frequently used in making business decisions.
• Different people often arrive at different
  subjective probabilities.

5
Fundamentals of Probability Distributions

• Frequency Distribution
• organization of numerical data about the events
• Probability Distribution
• A list of corresponding probabilities for each
  event
• Mutually Exclusive Events
• If two or more events cannot occur at the same
  time
• Probability that one or more events will occur is
  found by summing the individual probabilities of
  the events
• P(A or B) P(A) P(B)

6
Fundamentals of Probability A Frequency
Distribution Example

• Grades for past four years.

7
Fundamentals of Probability Non-Mutually
Exclusive Events Joint Probability

• Probability that non-mutually exclusive events M
  and F or both will occur expressed as
• P(M or F) P(M) P(F) - P(MF)
• A joint (intersection) probability, P(MF), is the
  probability that two or more events that are not
  mutually exclusive can occur simultaneously.

8
Fundamentals of Probability Cumulative
Probability Distribution

• Determined by adding the probability of an event
  to the sum of all previously listed probabilities
• Probability that a student will get a grade of C
  or higher
• P(A or B or C) P(A) P(B) P(C) .10 .20
  .50 .80

9
Statistical Independence and Dependence Independen
t Events

• Events that do not affect each other are
  independent.
• Computed by multiplying the probabilities of each
  event.
• P(AB) P(A) ? P(B)
• For coin tossed three consecutive times
  Probability of getting head on first toss, tail
  on second, tail on third is
• P(HTT) P(H) ? P(T) ? P(T) (.5)(.5)(.5) .125

10
Statistical Independence and Dependence Independen
t Events Bernoulli Process Definition

• Properties of a Bernoulli Process
• Two possible outcomes for each trial.
• Probability of the outcome remains constant over
  time.
• Outcomes of the trials are independent.
• Number of trials is discrete and integer.

11
Binomial Distribution

• Used to determine the probability of a number of
  successes in n trials.
• where p probability of a success
• q 1- p probability of a failure
• n number of trials
• r number of successes in n trials
• Determine probability of getting exactly two
  tails in three tosses of a coin.

12
Example

• Microchips are inspected at the quality control
  station
• From every batch, four are selected and tested
  for defects
• Given defective rate of 20, what is the
  probability that each batch contains exactly two
  defectives

13
Binomial Distribution Example Quality Control

• What is probability that each batch will contain
  exactly two defectives?
• What is probability of getting two or more
  defectives?
• Probability of less than two defectives
• P(rlt2) P(r0) P(r1) 1.0 - P(r2)
  P(r3) P(r4)
• 1.0 - .1808 .8192

14
Dependent Events

• If the occurrence of one event affects the
  probability of the occurrence of another event,
  the events are dependent.
• Coin toss to select bucket, draw for blue ball.
• If tail occurs, 1/6 chance of drawing blue ball
  from bucket 2 if head results, no possibility of
  drawing blue ball from bucket 1.
• Probability of event drawing a blue ball
  dependent on event flipping a coin.

15
Dependent Events Conditional Probabilities

• Unconditional P(H) .5 P(T) .5, must sum to
  one.
• Conditional P(R?H) .33, P(W?H) .67, P(R?T)
  .83, P(W?T) .17

16
Math Formulation of Conditional Probabilities

• Given two dependent events A and B
• P(A?B) P(AB)/P(B) or P(AB) P(AB).P(B)
• With data from previous example
• P(RH) P(R?H) ? P(H) (.33)(.5) .165
• P(WH) P(W?H) ? P(H) (.67)(.5) .335
• P(RT) P(R?T) ? P(T) (.83)(.5) .415
• P(WT) P(W?T) ? P(T) (.17)(.5) .085

17
Summary of Example Problem Probabilities
18
Bayesian Analysis

• In Bayesian analysis, additional information is
  used to alter (improve) the marginal probability
  of the occurrence of an event.
• Improved probability is called posterior
  probability
• A posterior probability is the altered marginal
  probability of an event based on additional
  information.
• Bayes Rule for two events, A and B, and third
  event, C, conditionally dependent on A and B

19
Bayesian Analysis Example (1 of 2)

• Machine setup if correct 10 chance of defective
  part if incorrect, 40.
• 50 chance setup will be correct or incorrect.
• What is probability that machine setup is
  incorrect if sample part is defective?
• Solution P(C) .50, P(IC) .50, P(DC) .10,
  P(DIC) .40
• where C correct, IC incorrect, D defective

20
Statistical Independence and Dependence Bayesian
Analysis Example (2 of 2)

• Previously, the manager knew that there was a 50
  chance that the machine was set up incorrectly
• Now, after testing the part, he knows that if it
  is defective, there is 0.8 probability that the
  machine was set up incorrectly

21
Expected Value Random Variables

• When the values of variables occur in no
  particular order or sequence, the variables are
  referred to as random variables.
• Random variables are represented by a letter x,
  y, z, etc.
• Possible to assign a probability to the
  occurrence of possible values.

Possible values of no. of heads are
Possible values of demand/week
22
Expected Value Example (1 of 4)

• Machines break down 0, 1, 2, 3, or 4 times per
  month.
• Relative frequency of breakdowns , or a
  probability distribution

23
Expected Value Example (2 of 4)

• Computed by multiplying each possible value of
  the variable by its probability and summing these
  products.
• The weighted average, or mean, of the probability
  distribution of the random variable.
• Expected value of number of breakdowns per month
• E(x) (0)(.10) (1)(.20) (2)(.30)
  (3)(.25) (4)(.15)
• 0 .20 .60 .75 .60
• 2.15 breakdowns

24
Expected Value Example (3 of 4)

• Variance is a measure of the dispersion of random
  variable values about the mean.
• Variance computed as follows
• Square the difference between each value and
  the expected value.
• Multiply resulting amounts by the probability
  of each value.
• Sum the values compiled in step 2.
• General formula
• ?2 ?xi - E(xi) 2 P(xi)

25
Expected Value Example (4 of 4)

• Standard deviation computed by taking the square
  root of the variance.
• For example data
• ?2 1.425 breakdowns per month
• standard deviation ? sqrt(1.425)
• 1.19 breakdowns per month

26
Poisson Distribution

• Based on the number of outcomes occurring during
  a given time interval or in a specified regions
• Examples
• of accidents that occur on a given highway
  during a 1-week period
• of customers coming to a bank during a 1-hour
  interval
• of TVs sold at a department store during a
  given week
• of breakdowns of a washing machine per month

27
Conditions

• Consider the of breakdowns of a washing machine
  per month example
• Each breakdown is called an occurrence
• Occurrences are random that they do not follow
  any pattern (unpredictable)
• Occurrence is always considered with respect to
  an interval (one month)

28
The Probability Mass Distribution

• X number of counts in the interval
• Poisson random variable with ? gt 0
• PMF
• f(x) x0,1,2,?
• Mean and Variance
• EX ? , V (X) ?

29
Example

• If a bank gets on average ? 6 bad checks per
  day, what are the probabilities that it will
  receive four bad checks on any given day?10 bad
  checks on any two consecutive days?
• Solution
• x 4 and ? 6, then f(4)
  0.135
• ? 12 and x 10, then f(10)
  0.105

30
Example

• The number of failures of a testing instrument
  from contamination particle on the product is a
  Poisson random variable with a mean of 0.02
  failure per hour.
• What is the probability that the instrument does
  not fail in an 8-hour shift?
• What is the probability of at least one failure
  in one 24-hour day?

31
Solution

• Let X denote the failure in 8 hours. Then, X has
  a Poisson distribution with ?0.16
• P(X0)0.8521
• Let Y denote the number of failure in 24 hours.
  Then, Y has a Poisson distribution with ?0.48
• P(Y?1) 1-P(Y 0) 0.3812

32
The Normal Distribution Continuous Random
Variables

• Continuous random variable can take on an
  infinite number of values within some interval.
• Continuous random variables have values that are
  not countable
• Cannot assign a unique probability to each value

33
The Normal Distribution Definition

• The normal distribution is a continuous
  probability distribution that is symmetrical on
  both sides of the mean.
• The center of a normal distribution is its mean
  ?.
• The area under the normal curve represents
  probability, and total area under the curve sums
  to one.

34
The Normal Distribution Example (1 of 5)

• Mean weekly carpet sales of 4,200 yards, with
  standard deviation of 1,400 yards.
• What is probability of sales exceeding 6,000
  yards?
• ? 4,200 yd ? 1,400 yd probability that
  number of yards of carpet will be equal to or
  greater than 6,000 expressed as P(x?6,000).

35
The Normal Distribution Example (2 of 5)
- -
36
The Normal Distribution Standard Normal Curve (1
of 2)

• The area or probability under a normal curve is
  measured by determining the number of standard
  deviations from the mean.
• Number of standard deviations a value is from the
  mean designated as Z.
• Z (x - ?)/?

37
The Normal Distribution Standard Normal Curve (2
of 2)
38
The Normal Distribution Example (3 of 5)
Z (x - ?)/ ? (6,000 - 4,200)/1,400
1.29 standard deviations P(x? 6,000) .5000 -
.4015 .0985

39
The Normal Distribution Example (4 of 5)

• Determine probability that demand will be 5,000
  yards or less.
• Z (x - ?)/? (5,000 - 4,200)/1,400 .57
  standard deviations
• P(x? 5,000) .5000 .2157 .7157

40
The Normal Distribution Example (5 of 5)

• Determine probability that demand will be between
  3,000 yards and 5,000 yards.
• Z (3,000 - 4,200)/1,400 -1,200/1,400 -.86
• P(3,000 ? x ? 5,000) .2157 .3051 .5208

41
Different Table

• P(3,000 ? x ? 5,000)
• P((3,000 - 4,200)/1,400) ? z ? ((5,000 -
  4,200)/1,400)
• P(-0.86? z ? 0.57)
• P( z ? 0.57)- P( z ? -0.86)
• P( z ? 0.57)- P( z 0.86)
• P( z ? 0.57)- 1-P( z ? 0.86)
• (0.7157)-1-0.80510.5208

42
The Normal Distribution Sample Mean and Variance

• The population mean and variance are for the
  entire set of data being analyzed.
• The sample mean and variance are derived from a
  subset of the population data and are used to
  make inferences about the population.

43
The Normal Distribution Computing the Sample Mean
and Variance
44
The Normal Distribution Example Problem Re-Done
Sample mean 42,000/10 4,200 yd Sample
variance (190,060,000) - (1,764,000,000/10)/9
1,517,777 Sample std. dev.
sqrt(1,517,777)
1,232 yd
45
The Normal Distribution Chi-Square Test for
Normality (1 of 2)

• It can never be simply assumed that data are
  normally distributed.
• A statistical test must be performed to determine
  the exact distribution.
• The Chi-square test is used to determine if a set
  of data fit a particular distribution.
• It compares an observed frequency distribution
  with a theoretical frequency distribution that
  would be expected to occur if the data followed a
  particular distribution (testing the
  goodness-of-fit).

46
The Normal Distribution Chi-Square Test for
Normality (2 of 2)

• In the test, the actual number of frequencies in
  each range of frequency distribution is compared
  to the theoretical frequencies that should occur
  in each range if the data follow a particular
  distribution.
• A Chi-square statistic is then calculated and
  compared to a number, called a critical value,
  from a chi-square table.
• If the test statistic is greater than the
  critical value, the distribution does not follow
  the distribution being tested if it is less, the
  distribution does exist.
• Chi-square test is a form of hypothesis testing.

47
Statistical Analysis with Excel (1 of 3)
48
Statistical Analysis with Excel (2 of 3)
49
Statistical Analysis with Excel (3 of 3)