Defining Probabilities: Random Variables

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Defining Probabilities Random Variables

• Examples
• Out of 100 heart catheterization procedures
  performed at a local hospital each year, the
  probability that more than five of them will
  result in complications is
• __________
• Drywall anchors are sold in packs of 50 at the
  local hardware store. The probability that no
  more than 3 will be defective is
• __________
• In general, ___________

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Discrete Random Variables

• Example
• Look back at problem 3, page 46. Assume someone
  spends 75 to buy 3 envelopes. The sample space
  describing the presence of 10 bills (H) vs bills
  that are not 10 (N) is
• _____________________________
• The random variable associated with this
  situation, X, reflects the outcome of the choice
  and can take on the values
• _____________________________

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Discrete Probability Distributions

• The probability that there are no 10 in the
  group is
• P(X 0) ___________________
• (recall results from last time)
• The probability distribution associated with the
  number of 10 bills is given by

x 0 1 2 3
P(X x)
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Another Example

• Example 3.3, pg 66
• P(X 0)
• _____________________

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Discrete Probability Distributions

• The discrete probability distribution function
  (pdf)
• f(x) P(X x) 0
• Sx f(x) 1
• The cumulative distribution, F(x)
• F(x) P(X x) St x f(t)

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Probability Distributions

• From our example, the probability that no more
  than 2 of the envelopes contain 10 bills is
• P(X 2) F(2) _________________
• The probability that no fewer than 2 envelopes
  contain 10 bills is
• P(X 2) 1 - P(X 1) 1 - F(1)
  ________________

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Another View

• The probability histogram

8
Your Turn

• The output of the same type of circuit board from
  two assembly lines is mixed into one storage
  tray. In a tray of 10 circuit boards, 6 are from
  line A and 4 from line B. If the inspector
  chooses 2 boards from the tray, show the
  probability distribution function associated with
  the selected boards being from line A.

x P(x)
0 1 2
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Continuous Probability Distributions

• Examples
• The probability that the average daily
  temperature in Georgia during the month of August
  falls between 90 and 95 degrees is
• __________
• The probability that a given part will fail
  before 1000 hours of use is
• __________
• In general, __________

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Understanding Continuous Distributions

• The probability that the average daily
  temperature in Georgia during the month of August
  falls between 90 and 95 degrees is
• The probability that a given part will fail
  before 1000 hours of use is

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Continuous Probability Distributions

• The continuous probability density function (pdf)
  
• f(x) 0, for all x ? R
• The cumulative distribution, F(x)

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Probability Distributions

• Example Problem 7, pg. 73
• x, 0 lt x lt 1
• f(x) 2-x, 1 x lt 2
• 0, elsewhere
• 1st what does the function look like?
• P(X lt 120) ___________________
• P(50 lt X lt 100) ___________________

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Your turn

• Problem 14, pg. 73

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Additional useful information

• Joint probability distributions
• Example 1 (discrete) the joint probability mass
  function (see defn. 3.8, pg. 75)
• In the development of a new receiver for a
  digital communication system, each received bit
  is rated as acceptable, suspect, or unacceptable,
  depending on the quality of the received signal,
  with the following probabilities
• P(acceptable) P(x) 0.9
• P(suspect) P(y) 0.08
• P(unacceptable) P(z) 0.02

Section 3.4 in your book (Optional)
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• If we let X denote the number of acceptable bits
  and Y denote the number of suspect bits, then the
  joint probability associated with the number of
  acceptable and suspect bits in 4 transmitted bits
  is denoted by
• fXY(x, y), where fXY(x, y) 0
• ?x ?y fXY(x, y) 1
• fXY(x, y) P(X x, Y y)
• So, the probability of exactly two acceptable
  bits and exactly 1 suspect bit in the first 4
  bits is
• fXY(2, 1) P(X 2, Y 1)
• Assuming independence, we can determine the
  probability of a particular combination, say
  aasu, as
• P(aasu) P(a)P(a)P(s)P(u)
  0.90.90.080.02
• 0.0013

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• Recognizing that aasu is just one of several
  possible combinations of 4 bits, we next need to
  determine the number of possible permutations of
  2 acceptable and 1 suspect bit in 4 tested bits.
  That is (from theorem 2.6, pg. 36),
• So,
• fXY(2, 1) P(X 2, Y 1) 12(0.0013)
  0.0156

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Joint probability distributions

• Example 2 (continuous) the joint density
  function (see definition 3.9, pg. 76)
• If X denotes the time until a computer server
  connects to your machine (in milliseconds) and Y
  denotes the time until the server authorizes you
  as a valid user (in milliseconds.) Each measures
  the wait from a common starting time and X lt Y.
  Assume that the joint probability density
  function for X and Y is given as
• fXY(x, y) 6 x 10-6exp(-0.001x 0.002y) for x
  lt y

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• The probability that X lt 1000 and Y lt 2000 is

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• Note we can verify that this density function
  integrates to 1 as follows

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For further study

• Read section 3.4
• Solve selected problems on pp. 84-86