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Title: Probability%20Review-1


1
Probability Review
2
Probability Theory
  • Mathematical description of relationships or
    occurrences that cannot be predicted precisely
  • An experiment is an activity whose outcome is
    subject to random (i.e. chance or unknown)
    variation.Examples
  • Flip a coin
  • Toss a die

3
Sample Space
  • The set of all possible outcomes of an experiment
    is known as its sample space, denoted as
    SExamples
  • Coin flippingS
  • Tossing a die S

4
Events
  • An event is a collection of outcomes from a
    sample space, denoted as EExamples
  • Coin flipping Event Get a tail
  • Tossing a die Event Get an even number
    outcome 2,4,6
  • An event E is said to occur if one of the
    outcomes with which it is associated is realized
    during a replication of the experiment

5
Events
  • Two events, E and F, are said to be mutually
    exclusive if E n F ? which means
  • The complement of event E, denoted Ec, is that
    unique set such that E U Ec S and E n Ec ?

6
Random Variable
  • A random variable is a function that maps the
    sample space to the real line.Examples
  • Coin flippingX 1 if heads 0 if tails
  • Tossing a die W (the number that shows on die)
  • A random variable is discrete if the possible
    values it can assume can be counted
  • A random variable is continuous if it can assume
    any value in a continuous subset of the real line

7
Probability
  • The probability associated with a particular
    event E, denoted P(E), can be thought of as
    representing the relative likelihood of that
    event occurring
  • We will be generally thinking in terms of the
    probability of a random variable taking a
    specific value Examples
  • Coin flippingP(X1)
  • Tossing a die P(W6) P(W7)

8
Axioms of Probability
  • 0 P(E) 1 for any E
  • P(S) 1
  • If Ei, i1,,k are mutually exclusive events,
    then

9
Probability Distributions
  • Describes probabilities of values a random
    variable could take
  • DiscreteExamples
  • Coin flippingP(Xx) ½ if x0,1 0
    otherwise
  • Tossing a die P(Ww) 1/6 if w1,2,3,4,5,6
    0 otherwise
  • ContinuousExamples
  • Altitude of an airplaneArea under curve

10
Common Probability Distributions
  • Discrete
  • Discrete uniform
  • Poisson
  • Geometric
  • Binomial
  • Continuous
  • Uniform
  • Exponential
  • Normal
  • Gamma
  • Beta
  • Triangular

11
PDF(PMF) vs. CDF
  • Probability density function (p.d.f.) denote f
  • Probability mass function (p.m.f.) denote f
  • f(x)P(Xx)
  • Cumulative distribution function (c.d.f.) denote
    FF(x)P(Xx)

12
Mean and Variance
  • Mean (Expected Value)
  • EX ?

or
  • Variance (Expected square distance from mean)
  • Var(X) ?2 E(X-EX)2 EX2 EX2
  • Standard deviation (Spread)

13
Examples of Mean and Variance
  • Coin flipping
  • Expected value
  • Variance
  • Standard deviation
  • Tossing a die
  • Expected value
  • Variance
  • Standard deviation
  • Continuous uniform between 0 and 2
  • Expected value
  • Variance
  • Standard deviation

14
Conditional Probabilities
  • Consider two experiments with S1E1,,Em and
    S2F1,,Fn
  • P(EF) Pexperiment 1 gets outcome E given that
    experiment 2 gets outcome F
  • ExampleP(Ice cream sales gt 10 cones
    temperature 85 F)

15
Example 1 of Conditional Probabilities
  • The king comes from a family of 2 children. What
    is the probability that the other child is his
    sister?

16
Example 2 of Conditional Probabilities
  • 52 of the students at a certain college are
    females. 5 of the students in this college are
    majoring in computer science. 2 of the students
    are women majoring in computer science. If a
    student is selected at random, find the
    conditional probability that
  • this student is female, given that the student is
    majoring in computer science
  • this student is majoring in computer science,
    given that the student is female.

17
Bayes Theorem
18
Example 1 of Bayes Theorem
  • Suppose that an insurance company classifies
    people into one of three classes good risks,
    average risks, and bad risks. Their records
    indicate that the probabilities that good,
    average, and bad risk persons will be involved in
    an accident over a 1-year span are, respectively,
    0.05, 0.15, and 0.30. If 20 of the population
    are good risks, 50 are average risks, and
    30 are bad risks, what proportion of people
    have accidents in a fixed year? If policy holder
    A had no accidents in 1987, what is the
    probability that he or she is a good risk?

19
Example 2 of Bayes Theorem
  • Suppose that there was a cancer diagnostic test
    that was 95 accurate both on those that do and
    those that do not have the disease. If 0.4 of
    the population have a cancer, compute the
    probability that a tested person has cancer,
    given that his or her test result indicates so.
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