Topic III: Random Variables and Probability Distributions

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Topic III Random Variables and Probability
Distributions

• Discrete Random Variables and Probability
  Distributions

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Discrete Random Variables

• Can only take on a countable number of values
• Examples
• Roll a die twice
• Let X be the number of times 4 comes up
• (then X could be 0, 1, or 2 times)
• Toss a coin 5 times.
• Let X be the number of heads
• (then X 0, 1, 2, 3, 4, or 5)

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Discrete Probability Distributions
Experiment Toss 2 Coins. Let X heads.
Show P(x) , i.e., P(X x) , for all values
of x
4 possible outcomes
Probability Distribution

• x Value Probability
• 0 1/4 .25
• 1 2/4 .50
• 2 1/4 .25

T
T
T
H
H
T
.50 .25
Probability
H
H
0 1 2 x
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Probability Distribution Function

• The Probability Distribution Function (PDF),
  P(x), of a discrete random variable X expresses
  the probability that X takes the value x, as a
  function of x. That is
• P(x) P(Xx), for all values of x.

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Probability DistributionRequired Properties

• P(x) ? 0 for any value of x
• The individual probabilities sum to 1
• (The notation indicates summation over all
  possible x values)

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Cumulative Probability Function

• The cumulative probability function, denoted
  F(x0), shows the probability that X is less
  than or equal to x0
• In other words,

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Properties of Cumulative Probability Functions

• 0 ? F(x0) ? 1 for every number x0
• If x0 and x1 are two numbers with x0 ? x1, then
  F(x0) ? F(x1)

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Expected Value

• Expected Value (or mean) of a discrete
• distribution (Weighted Average)
• Example Toss 2 coins,
• x of heads,
• compute expected value of x
• E(x) (0 x .25) (1 x .50) (2 x .25)
• 1.0

x P(x) 0
.25 1 .50
2 .25
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Variance and Standard Deviation

• Variance of a discrete random variable X
• Standard Deviation of a discrete random variable
  X

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Standard Deviation Example

• Example Toss 2 coins, X heads,
• compute standard deviation (recall E(x) 1)

Possible number of heads 0, 1, or 2
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Functions of Random Variables

• If P(x) is the probability function of a
  discrete random variable X , and g(X) is some
  function of X , then the expected value of
  function g is

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Linear Functions of Random Variables

• Let a and b be any constants.
• a)
• i.e., if a random variable always takes the
  value a, it will have mean a and variance 0
• b)
• i.e., the expected value of bX is bE(x)

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Linear Functions of Random Variables
(continued)

• Let random variable X have mean µx and variance
  s2x
• Let a and b be any constants.
• Let Y a bX
• Then the mean and variance of Y are
• so that the standard deviation of Y is

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Probability Distributions
Probability Distributions
Continuous Probability Distributions
Discrete Probability Distributions
Binomial
Uniform
Hypergeometric
Normal
Poisson
Exponential
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The Binomial Distribution
Probability Distributions
Discrete Probability Distributions
Binomial
Hypergeometric
Poisson
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Bernoulli Distribution

• Consider only two outcomes success or
  failure
• Let P denote the probability of success
• Let 1 P be the probability of failure
• Define random variable X
• x 1 if success, x 0 if failure
• Then the Bernoulli probability function is

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Bernoulli DistributionMean and Variance

• The mean is µ P
• The variance is s2 P(1 P)

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Sequences of x Successes in n Trials

• The number of sequences with x successes in n
  independent trials is
• Where n! n(n 1)(n 2) . . . 1 and
  0! 1
• These sequences are mutually exclusive, since no
  two can occur at the same time

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Binomial Probability Distribution

• A fixed number of observations, n
• e.g., 15 tosses of a coin ten light bulbs taken
  from a warehouse
• Two mutually exclusive and collectively
  exhaustive categories
• e.g., head or tail in each toss of a coin
  defective or not defective light bulb
• Generally called success and failure
• Probability of success is P , probability of
  failure is 1 P

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Binomial Probability Distribution

• Constant probability for each observation
• e.g., Probability of getting a tail is the same
  each time we toss the coin
• Observations are independent
• The outcome of one observation does not affect
  the outcome of the other
• The distribution of the number of successes x
  resulting is called the binomial distribution.

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Possible Binomial Distribution Settings

• A manufacturing plant labels items as either
  defective or acceptable
• A firm bidding for contracts will either get a
  contract or not
• A marketing research firm receives survey
  responses of yes I will buy or no I will not
• New job applicants either accept the offer or
  reject it

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Binomial Distribution Formula
n
!
X
X
-
n
P(x)
P
(1- P)

)
x !
n
x
(
!
-
P(x) probability of x successes in n trials,
with probability of success P on each trial
x number of successes in sample,
(x 0, 1, 2, ..., n) n sample size
(number of trials or observations) P
probability of success
Example Flip a coin four times, let x
heads n 4 P 0.5 1 - P (1 - 0.5) 0.5 x
0, 1, 2, 3, 4
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Example Calculating a Binomial Probability
What is the probability of one success in five
observations if the probability of success is
0.1? x 1, n 5, and P 0.1
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Binomial Distribution

• The shape of the binomial distribution depends on
  the values of P and n

Mean
n 5 P 0.1
P(x)
.6

• Here, n 5 and P 0.1

.4
.2
0
x
0
1
2
3
4
5
n 5 P 0.5
P(x)
.6

• Here, n 5 and P 0.5

.4
.2
x
0
0
1
2
3
4
5
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Binomial DistributionMean and Variance

• Mean

• Variance and Standard Deviation

Where n sample size P probability of
success (1 P) probability of failure
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Binomial Characteristics
Examples
n 5 P 0.1
P(x)
Mean
.6
.4
.2
0
x
0
1
2
3
4
5
n 5 P 0.5
P(x)
.6
.4
.2
x
0
0
1
2
3
4
5
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Using Binomial Tables
Examples n 10, x 3, P 0.35 P(x
3n 10, p 0.35) .2522 n 10, x 8, P
0.45 P(x 8n 10, p 0.45) .0229
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Binomial Example 1

• Suppose that items coming off an assembly line
  are classified as defective (D) or non-defective
  (N). Suppose further that any item has a p(D)
  0.2 and all items are independent.
• If the random variable X takes on the value 0,
  if an item is non-defective and 1 if an item is
  defective then P(X1) p 0.2 and P(X0) 1-p
  0.8 and the random variable X is said to have a
  Bernoulli Distribution with p.2.
• If 3 items are chosen at random from the days
  production, and X defective, what is the PDF
  of X?

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Binomial Example 2

• A politician believes that 25 of all
  macroeconomists in senior positions would support
  a proposal he wishes to advance. Suppose that
  this belief is correct and that five senior
  macroeconomists are approached at random.
• What is the probability that at least one of the
  five would strongly support the proposal?
• What is the probability that a majority of the
  five would strongly support the proposal?

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The Hypergeometric Distribution
Probability Distributions
Discrete Probability Distributions
Binomial
Hypergeometric
Poisson
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The Hypergeometric Distribution

• Assume we are playing a card game with a regular
  deck of 52 cards, where 16 of these are face
  cards'' and each hand'' consists of 10 randomly
  selected cards. Using combinations, find the
  probability of getting 4 face cards in a hand of
  10 cards. We know there are possible
  hands'' and possible hands with 4
  face cards, therefore

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The Hypergeometric Distribution

• n trials in a sample taken from a finite
  population of size N
• Sample taken without replacement, hence the
  probability of a success changes with each trial
• Outcomes of trials are dependent
• Concerned with finding the probability of X
  successes in the sample where there are S
  successes in the population

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Hypergeometric Distribution Formula
Where N population size S number of
successes in the population N S number
of failures in the population n sample size x
number of successes in the sample n x
number of failures in the sample
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Using the Hypergeometric Distribution

• Example 3 different computers are checked from
  10 in the department. 4 of the 10 computers have
  illegal software loaded. What is the probability
  that 2 of the 3 selected computers have illegal
  software loaded?
• N 10 n 3
• S 4 x 2

The probability that 2 of the 3 selected
computers have illegal software loaded is 0.30,
or 30.
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Hypergeometric Example 1

• We pick 3 people at random from a group of 50
  people, in which 20 of them have traded stocks on
  the Internet.
• Let X people in our sample who have
  traded stocks on the Internet. x 0,1,2,3
• What is the PDF of the discrete random variable
  X?

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Hypergeometric Example 2

• A company receives a shipment of sixteen items. A
  random sample of four items is selected, and the
  shipment is rejected if any of these items proves
  to be defective.
• What is the probability of accepting a shipment
  containing four defective items?
• What is the probability of accepting a shipment
  containing two defective items?
• What is the probability of rejecting a shipment
  containing one defective item?

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The Poisson Distribution
Probability Distributions
Discrete Probability Distributions
Binomial
Hypergeometric
Poisson
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The Poisson Distribution

• Apply the Poisson Distribution when
• You wish to count the number of times an event
  occurs in a given continuous interval
• The probability that an event occurs in one
  subinterval is very small and is the same for all
  subintervals
• The number of events that occur in one
  subinterval is independent of the number of
  events that occur in the other subintervals
• There can be no more than one occurrence in each
  subinterval
• The average number of events per interval is ?
  (lambda)

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Poisson Distribution Formula
where x number of successes per interval ?
expected number of successes per interval e
base of the natural logarithm system
(2.71828...)
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The Poisson Distribution

• The Poisson probability distribution is an
  important discrete probability distribution for a
  number of applications, including
• The number of failures in a large computer system
  during a given day
• The number of delivery trucks to arrive at a
  central warehouse in an hour
• The number of customers to arrive for flights
  during each 15-minute time interval from 300 PM
  to 600 PM on weekdays
• The number of customers to arrive at a checkout
  aisle in your local grocery store during a
  particular time interval.

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Poisson Distribution Characteristics

• Mean

• Variance and Standard Deviation

where ? expected number of successes per unit
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Using Poisson Tables
Example Find P(X 2) if ? .50
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Graph of Poisson Probabilities
Graphically
? .50
P(X 2) .0758
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Poisson Distribution Shape

• The shape of the Poisson Distribution depends on
  the parameter ?

? 0.50
? 3.00
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Poisson Distribution Shape
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Poisson- Example 1

• Customers arrive at a busy check-out counter at
  an average rate of three per minute. If the
  distribution of arrivals is Poisson, find the
  probability that in any given minute there will
  be two or fewer arrivals.

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Poisson- Example 2

• A professor receives, on average, 4.3 telephone
  calls from students the day before a final
  examination. If the distribution of calls is
  Poisson, what is the probability of receiving at
  least three of these calls on such a day?

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Poisson Approximation to the Binomial Distribution

• Let X be the number of successes resulting from n
  independent trials, each with probability of
  success, p. The distribution of the number of
  successes X is binomial, with mean np. If the
  number of trials n is large and np is of only
  moderate size (preferably np ? 7), this
  distribution can be approximated by the Poisson
  distribution with The probability
  function of the approximating distribution is
  then
• 
  for x 0,1,2,

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Poisson Approximation to the Binomial Distribution

• Example 1
• Binomial situation, n 100, p0.065
• Calculate the probability of fewer than 10
  successes.

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Poisson Approximation to the Binomial Distribution

• Example 2
• The Inland Revenue Department reported that 5.5
  of all taxpayers filling out the P45 form make
  mistakes. If 100 of these forms are chosen at
  random, what is the probability that fewer than
  three of them contain errors? Use the Poisson
  approximation to the binomial distribution.