Todays Goals

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Todays Goals

• Review
• Mini-Project due May 15.
• Sample final and solutions are posted.
• Recommended practice problems 6.9 7.13, 7.33c,
  7.35a, 7.37a 8.29a, 8.31, 8.35, 8.53 9.3

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Example of article HW

• TV causes depression in teens
• Alt depressed teens watch more TV
• Earlier start time in school, more accidents
• inference lack of sleep is dangerous
• alt inference more cars/traffic at new start
  time
• Participants lost 18lbs eating Healthy Choice
• problem no control group!
• Optimists live longer than pessimists
• Alt healthier people are most optimistic

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Topics on Final

• Calculating Probabilities
• Using independence
• Mutually exclusive
• Conditional probability
• Flaw of Averages (Single and multiple variables)
• Applying Bayes Rule and Total Probability

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Topics on Final

• Applying Probability Models
• Discrete Models
• Binomial
• HyperGeometric
• Negative Binomial
• Poisson
• Continuous
• Uniform
• Normal
• Exponential

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Topics on Final

• Joint Probability
• Covariance Correlation
• Joint probability Distributions
• Means of a function of variables (flaw of
  averages)
• The distribution of sample means
• Confidence Intervals
• Hypothesis Testing

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Calculating Probabilities

• If two events are independent, then
• p(x and y) p(x) p(y)
• If events are mutually exclusive (and exhaustive)
  then
• p(a) p(b) 1
• The probability that a doesnt happen is
• p(-a) 1-p(a)

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Example

• A vehicle contains two engines, a main engine and
  a backup. The engine component fails only if both
  engines fail. The probability that the main
  engine fails is 0.05, and the probability that
  the backup engine fails is 0.10. Assume that the
  main and backup engines function independently.
  What is the probability that the engine component
  does not fail?

A. .995 B. .95 C. .9 D.
.855
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Example

• A vehicle contains two engines, a main engine and
  a backup. The engine component fails only if both
  engines fail. The probability that the main
  engine fails is 0.05, and the probability that
  the backup engine fails is 0.10. Assume that the
  main and backup engines function independently.
  What is the probability that the engine component
  does not fail?
• p(fails)p(both engines fail) .05 .1 .005
• p(not fails) 1-p(fails) 1-.005 .995

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Conditional Probability

• P(AB) denotes the probability of event A
  occurring given event B occurs.

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Multiplication Rule

• As a consequence we have that,
• Example In a multiple choice test with 3 choices
  per question,
• a student that does not know the answer will
  guess right with probability 1/3.
• Suppose the probability of knowing the answer is
  2/3,
• what is the probability of not knowing the answer
  and answering correctly?

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Law of Total Probability

• If a student passed the midterm there is a 90
  chance they will pass the final.
• If they failed the midterm, there is a 30 chance
  they will fail the final.
• If 20 of the class failed the midterm, what is
  the probability that a randomly selected student
  will fail the final?

A. 2 B. 6 C. 20 D. 14
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Law of Total Probability
If a student passed the midterm there is a 90
chance they will pass the final. If they failed
the midterm, there is a 30 chance they will fail
the final. If 20 of the class failed the
midterm, what is the probability that a randomly
selected student will fail the final? A fail
the final B fail the midterm P(AB)
0.3 P(AB) 0.1 P(B) 0.2
P(A) 0.30.2 0.10.8 0.14
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Bayes Rule
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Bayes Rule

• If a student passed the midterm there is a 90
  chance they will pass the final. If they failed
  the midterm, there is a 30 chance they will fail
  the final. 20 of the class failed the midterm.
• What is the probability that a student failed the
  midterm given that they failed the final?

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Bayes Rule

• IIf a student passed the midterm there is a 90
  chance they will pass the final. If they failed
  the midterm, there is a 30 chance they will fail
  the final. 20 of the class failed the midterm.
• What is the probability that a student failed the
  midterm given that they failed the final?
• A fail the final
• B fail the midterm

p(AB) 0.3 p(B) 0.2 p(A) 0.14
p(BA) (0.30.2)/.14 0.43
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Probability Models -- Discrete

• Binomial
• What is the probability of x successes out of n
  trials.
• The probability p of a success is unaffected by
  previous successes.
• Hypergeometric
• What is the probability of x successes out of n
  trials
• sampling without replacement the probability of
  a success depends on what happened before
• Negative Binomial
• What is the probability of x failures before r
  successes?
• the probability p of a success is unaffected by
  previous successes.
• Poisson
• What is the probability of x events within a
  given time period?

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Binomial Distribution

• If then
• E(X)  np,
• V(X)  np(1  p)  npq.

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Hypergeometric Distribution

• If X is the number of Ss in a completely random
  sample of size n drawn from a population
  consisting of M Ss and (NM) Fs, then the
  probability distribution of a hypergeometric
  distribution is given by

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Negative Binomial Distribution

• The pmf of a negative binomial rv X with
  parameters r  number of Ss and p  P(S) is
• If X is a negative binomial r.v. with pmf
  nb(xr,p), then

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Poisson Random Variable

• A random variable X is said to have a Poisson
  distribution if the pmf of X is
• If X has a Poisson distribution with parameter l,
  then

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Example

• I test 10 parts out of a large lot. The
  manufacturer claims that they have a 20 defect
  rate. What is the probability that I will find 2
  or more defective parts?

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Example

• I test 10 parts out of a large lot. The
  manufacturer claims that they have a 20 defect
  rate. What is the probability that I will find 2
  or more defective parts?
• Binomial
• n 10
• p 20
• p(X 2) 1 - p(X0 or 1)
• 1 - .44 .56

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Probability Models - Continuous

• Uniform
• Normal
• Exponential

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Probability Density Function
is given by the area of the
shaded region.
Note It is the area under the p.d.f. curve that
has a probability interpretation- not the height
of the p.d.f. curve.

• Its graph, called the density or p.d.f. curve
  shows how the total probability of 1 is spread
  over the range of X.

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Uniform Distribution

• A continuous rv X is said to have a uniform
  distribution on the interval A, B if the pdf of
  X is

f(x)

• Intervals with the same size have the same
  probability associated.

1/(B-A)
x
A
B
R.v. X models a random point in the interval A,B
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Normal DistributionPractical and Theoretical
Importance

• The real world presents us with a number of
  situations, in fact most situations, where our
  measurements are normally distributed
• symmetric, bell shaped curve around the mean,
  which is also equal to the median and the mode
•  

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Normal Probability Calculations Standardization

• Then, we can use the standard tables

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Exponential Distribution

• where ?0 is a given constant.
• Often used to model
• The lifetime of an item or survival time of a
  patient.
• Time between the occurrence of successive
  events
• arrivals to a service facility,
• calls coming to a switchboard

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Exponential Distribution Example

• where ?0 is a given constant.
• What is p(X

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Exponential Distribution Example

• where ?0 is a given constant.
• What is p(X
• EX 1/l

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Exponential Distribution Example

• where ?0 is a given constant.
• What is p(X
• EX 1/l

Is the Exponential Distribution symmetric?