Random Variables and Probability Distributions chapter 3

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Session 2

• Random Variables and Probability Distributions
  (chapter 3)

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Probability defined

• Probability measures the likelihood that an event
  will occur
• But what is an event?

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Terms used

• Experiment the process in which observation is
  obtained
• Example Conduct market survey
• Outcome any possible distinct result of an
  experiment
• Example Proportion of respondents that favor a
  product
• Event - a collection of one or more outcomes of
  an experiment that we are interested in (event is
  defined by us)
• Example A at least 70 of respondents favor
  the product
• Another example
• Experiment roll 2 dice.
• Outcome sum of dots in a roll.
• Possible events we might be interested in A
  7 or more dots, B even number dots, C 8

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Probability three concepts

• Classical definition based on theory (a priori)
• Relative frequency based on empirical data (a
  posteriori). Known also as empirical probability
• Subjective based on one-shot judgment

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Classical Definition of Probability

• Based on theory all outcomes are known a priori
• Probability number of possible favorable
  outcomes divided by the total number of possible
  outcomes
• Example There are four kings in a deck of cards
  and 52 cards total. Therefore, the probability of
  drawing a king is
• 4/52 1/13 .077.

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Relative Frequency Definition

• Based on experience the outcomes are observed,
  and the favorable outcomes are counted.
• Probability number of favorable outcomes
  observed divided by the total number of
  observations.
• Example Weather conditions have been observed
  for ten days, and it has rained eight times.
  Therefore, the probability of rain the next day
  is
• 8/10 0.80

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Subjective Definition

• An individual judgment or opinion about the
  probability of occurrence.
• Examples
• What is the probability that the New York Yankees
  will win the World Series this year?
• What is the probability our competitors will
  match our price decrease?
• What is the probability the NASDAQ will go up 10
  next week?

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Terms used, part 2

• Events are collectively exhaustive if there is no
  other event possible (Ex male or female)
• Two events are mutually exclusive if the
  occurrence of any one means that none of the
  other can occur at the same time. (Ex male or
  female)

C and B are NOT mutually exclusive
A and B are mutually exclusive
A
B
C
B C
B
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Basic Probability Rules

• The probability of any event must be between 0
  and 1, inclusively
• The sum of probabilities of all mutually
  exclusive and collectively exhaustive events is
  always 1

Certain
1
0 P(A) 1 For any event A
0.5
If A, B, and C are mutually exclusive and
collectively exhaustive
Impossible
0
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Probability Rules again

• Probability of any event or outcome must be
  between 0 and 1.
• Probability of any event is the sum of the
  outcomes that compose that event. Example P(even
  sum of dots) P(2)P(4)P(6)
• Sum of probabilities of all possible outcomes
  must be 1.0
• Example one coin toss. Possible outcomes
• H or T. Probabilities
• P(H) ½ P(T) ½ . Their sum 1.

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Joint probability

• Probability that two or more events occur
  simultaneously P(A and B)
• Examples
• Probability that in a three-coin toss well
  obtain all heads P(HHH)
• Note these three events HHH, are independent
• Probability that of three random selected
  employees all will be female
• Note these three events FFF, are dependent

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Joint probability

• If events are independent, then their joint
  probability is a product of their separate
  probabilities
• P(A and B) P(A)P(B)
• Example P(HHH) P(H)P(H) )P(H)
• 1/21/21/2 1/8

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Joint probability

• If events are dependent, then its joint
  probability is a product of the probability of
  one event times a conditional probability of the
  other event(s)
• P(A and B) P(A)P(BA)
• Example (FFF) P(F)P(FF) )P(FFF)
• Assuming that of 50 employees 30 are females.
  Then P(FFF)
• 30/50 29/49 28/48 0.2071

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Probability Rules cont

• 4. If events A and B are mutually exclusive, then
  P(A or B) P(A) P(B)
• Example probability that in a 3-coin toss well
  get at least two heads is
• P(HH) P(HHH) (exercise will follow)
• 5. If events A and B are not mutually exclusive,
  then P(A or B) P(A) P(B) P(A and B)

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Random Variables and their Probability
Distributions

• Random Variable represents a possible numerical
  value from an uncertain event

Random Variables
Discrete Random Variable
Continuous Random Variable
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Discrete and Continuous Random Variables

• Discrete with a countable number of values
• Examples number of inaccurate orders, number of
  complaints per day, number of customers served in
  a day
• Continuous assumes a countless number of values
  in any interval
• Examples temperature, length of the nail
  manufactured, weight of soft drink in a 2-liter
  bottle, thickness of a rod, time to complete a
  task, height in inches, rate of return on
  investment

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Example (discrete variable)

• Experiment flip a coin 3 times.
• Outcomes
• TTT, TTH, THT, THH, HTT, HTH, HHT, HHH
• Random variable X number of heads.
• X can be either 0, 1, 2, or 3.

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Probability Distributions

• Probability distribution a table, formula or
  graph listing all possible values a random
  variable may assume along with the probabilities
  of occurrence.
• Depending on the variable, we have
• Discrete probability distributions,
• Continuous probability distributions.

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Discrete Random Distribution - Summary Measures

• Expected Value (or mean) of a discrete
• distribution (Weighted Average)

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Discrete Random Distributions - Summary Measures
(continued)

• Variance of a discrete random variable
• Standard Deviation of a discrete random variable
• where
• E(X) Expected value (mean) of the discrete
  random variable X
• Xi the ith outcome of X
• P(Xi) Probability of the ith occurrence of X

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Discrete Probability Distributions - examples

• Bernoulli
• Binomial
• Poisson

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Continuous distributions

• Normal distribution
• Standard Normal distribution
• Student t distribution
• Uniform distribution
• Triangular distribution

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Normal Distribution

• Familiar bell-shaped curve.
• Symmetric, median mean mode half the area is
  on either side of the mean
• Range is unbounded the curve never touches the
  x-axis Parameters
• Mean, m (location)
• Variance s2 gt 0 (scale)
• Density function

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Many Normal Distributions
By varying the parameters µ and s, we obtain
different normal distributions
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Standard Normal Distribution

• Standard normal mean 0, variance 1, denoted
  as N(0,1)
• See Table A.1
• Transformation from N(m,s) to N(0,1)

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Areas Under the Normal Curve

• About 68.3 is within one sigma of the mean
• About 95.4 is within two sigma of the mean
• About 99.7 is within three sigma of the mean

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Normal Probability Calculations

• Customer demand averages 750 units/month with a
  standard deviation of 100 units/month.
• Find
• P(Xgt900)
• P(Xgt 700),
• the level of demand that will be exceeded only
  10 of the time.

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P(Xgt900)
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P(Xgt700)
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P(Xgtx)0.10
First, find the value of z from Table A.1, then
solve for x
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Excel Support

• NORMDIST(x, mean, standard_deviation, cumulative)
• NORMSDIST(z) same as Table A.1
• NORMINV(probability, mean, st._deviation)
• STANDARDIZE(x, mean, standard_deviation) computes
  z-values

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Triangular Distribution

• Three parameters
• Minimum, a
• Maximum, b
• Most likely, c
• a is the location parameter
• (b a) the scale parameter,
• c the shape parameter.