Chapter 4: Discrete Random Variables

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Statistics

• Chapter 4 Discrete Random Variables

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Where Weve Been

• Using probability to make inferences about
  populations
• Measuring the reliability of the inferences

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Where Were Going

• Develop the notion of a random variable
• Numerical data and discrete random variables
• Discrete random variables and their probabilities

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4.1 Two Types of Random Variables

• A random variable is a variable hat assumes
  numerical values associated with the random
  outcome of an experiment, where one (and only
  one) numerical value is assigned to each sample
  point.

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4.1 Two Types of Random Variables

• A discrete random variable can assume a countable
  number of values.
• Number of steps to the top of the Eiffel Tower
• A continuous random variable can assume any value
  along a given interval of a number line.
• The time a tourist stays at the top
• once s/he gets there

Believe it or not, the answer ranges from 1,652
to 1,789. See Great Buildings
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4.1 Two Types of Random Variables

• Discrete random variables
• Number of sales
• Number of calls
• Shares of stock
• People in line
• Mistakes per page

• Continuous random variables
• Length
• Depth
• Volume
• Time
• Weight

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4.2 Probability Distributions for Discrete
Random Variables

• The probability distribution of a discrete random
  variable is a graph, table or formula that
  specifies the probability associated with each
  possible outcome the random variable can assume.
• p(x) 0 for all values of x
• ?p(x) 1

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4.2 Probability Distributions for Discrete
Random Variables

• Say a random variable x follows this pattern
  p(x) (.3)(.7)x-1
• for x gt 0.
• This table gives the probabilities (rounded to
  two digits) for x between 1 and 10.

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4.3 Expected Values of Discrete Random Variables

• The mean, or expected value, of a discrete random
  variable is

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4.3 Expected Values of Discrete Random Variables

• The variance of a discrete random variable x is
• The standard deviation of a discrete random
  variable x is

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4.3 Expected Values of Discrete Random Variables
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4.3 Expected Values of Discrete Random Variables

• In a roulette wheel in a U.S. casino, a 1 bet on
  even wins 1 if the ball falls on an even
  number (same for odd, or red, or black).
• The odds of winning this bet are 47.37

On average, bettors lose about a nickel for each
dollar they put down on a bet like this. (These
are the best bets for patrons.)
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4.4 The Binomial Distribution

• A Binomial Random Variable
• n identical trials
• Two outcomes Success or Failure
• P(S) p P(F) q 1 p
• Trials are independent
• x is the number of Successes in n trials

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4.4 The Binomial Distribution

• Flip a coin 3 times
• Outcomes are Heads or Tails
• P(H) .5 P(F) 1-.5 .5
• A head on flip i doesnt change P(H) of flip i
  1

• A Binomial Random Variable
• n identical trials
• Two outcomes Success or Failure
• P(S) p P(F) q 1 p
• Trials are independent
• x is the number of Ss in n trials

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4.4 The Binomial Distribution
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4.4 The Binomial Distribution

• The Binomial Probability Distribution
• p P(S) on a single trial
• q 1 p
• n number of trials
• x number of successes

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4.4 The Binomial Distribution

• The Binomial Probability Distribution

The probability of getting the required number
of successes
The probability of getting the required number
of failures
The number of ways of getting the desired results
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4.4 The Binomial Distribution

• Say 40 of the class is female.
• What is the probability that 6 of the first 10
  students walking in will be female?

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4.4 The Binomial Distribution

• A Binomial Random Variable has

• Mean
• Variance
• Standard Deviation

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4.4 The Binomial Distribution

• For 1,000 coin flips,

The actual probability of getting exactly 500
heads out of 1000 flips is just over 2.5, but
the probability of getting between 484 and 516
heads (that is, within one standard deviation of
the mean) is about 68.
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4.5 The Poisson Distribution

• Evaluates the probability of a (usually small)
  number of occurrences out of many opportunities
  in a
• Period of time
• Area
• Volume
• Weight
• Distance
• Other units of measurement

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4.5 The Poisson Distribution

• ? mean number of occurrences in the given unit
  of time, area, volume, etc.
• e 2.71828.
• µ ?
• ?2 ?

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4.5 The Poisson Distribution

• Say in a given stream there are an average of 3
  striped trout per 100 yards. What is the
  probability of seeing 5 striped trout in the next
  100 yards, assuming a Poisson distribution?

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4.5 The Poisson Distribution

• How about in the next 50 yards, assuming a
  Poisson distribution?
• Since the distance is only half as long, ? is
  only half as large.

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4.6 The Hypergeometric Distribution

• In the binomial situation, each trial was
  independent.
• Drawing cards from a deck and replacing the drawn
  card each time
• If the card is not replaced, each trial depends
  on the previous trial(s).
• The hypergeometric distribution can be used in
  this case.

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4.6 The Hypergeometric Distribution

• Randomly draw n elements from a set of N
  elements, without replacement. Assume there are
  r successes and N-r failures in the N elements.
• The hypergeometric random variable is the number
  of successes, x, drawn from the r available in
  the n selections.

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4.6 The Hypergeometric Distribution
where N the total number of elements r
number of successes in the N elements n number
of elements drawn X the number of successes in
the n elements
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4.6 The Hypergeometric Distribution
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4.6 The Hypergeometric Distribution

• Suppose a customer at a pet store wants to buy
  two hamsters for his daughter, but he wants two
  males or two females (i.e., he wants only two
  hamsters in a few months)
• If there are ten hamsters, five male and five
  female, what is the probability of drawing two of
  the same sex? (With hamsters, its virtually a
  random selection.)