6.9

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6.9 Discrete Random Variables

• IBHLY2 - Santowski

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(A) Random Variables

•  Now we wish to combine some basic statistics
  with some basic probability ? we are interested
  in the numbers that are associated with
  situations resulting from elements of chance i.e.
  in the values of random variables
• We also wish to know the probabilities with which
  these random variables take in the range of their
  possible values ? i.e. their probability
  distributions

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(A) Random Variables

• So 2 definitions need to be clarified
• (i) a discrete random variable is a variable
  quantity which occurs randomly in a given
  experiment and which can assume certain, well
  defined values, usually integral ? examples
  number of bicycles sold in a week, number of
  defective light bulbs in a shipment
• discrete random variables involve a count
•  
• (ii) a continuous random variable is a variable
  quantity which occurs randomly in a given
  experiment and which can assume all possible
  values within a specified range ? examples the
  heights of men in a basketball league, the volume
  of rainwater in a water tank in a month
• continuous random variables involve a measure

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(B) CLASSWORK

• CLASSWORK (to review the distinction between the
  2 types of random variables)
• Math SL text, pg 710, Chap29A, Q1,2,3
• Math HL Text, p 728, Chap 30A, Q1,2,3

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(C) The Distributions of Random Variables

• For any random variable, there is an associated
  probability distribution
•  
• the discrete probability distribution is
  associated with discrete random variables ? so
  this probability distribution describes a
  discrete random variable in terms of the
  probabilities associated with each individual
  value that the variable may take
• the normal distribution is associated with
  continuous random variables
• we will initially consider only discrete data and
  their associated probability distributions

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(C) The Distributions of Random Variables

• We will toss three coins. The random variable, X,
  will represent the number of heads obtained.
  Construct a table and a graph to represent the
  discrete probability distribution
• the probability of exactly 1 head in three tosses
  will be written as P(X1) (which I can read as
  the probability that my random variable ( of
  heads) has the value 1 i.e. 1 head is tossed)
• It can be determined in many different ways ? I
  will use binomial probability distribution ((p
  q)3) from our last section ? C(3,1) x (0.5)1 x
  (0.5)3-1 3 x 0.5 x 0.25 0.375
• Or I could use a GDC and determine
  binompdf(3,0.5,1) 0.375
• (Or I could use the Fundamental Counting
  Principle gt p(H) x p(T) x p(T) x
  C(3,1) ) 0.375
• Likewise, I could do similar calculations to find
  the associated probabilities for 0,1,2,3 Heads
  gt I will write this as P(X x) and equate it
  to C(3,x) x (0.5)x x (0.5)3-x, x 0,1,2,3

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(C) The Distributions of Random Variables

• I get the following graph

• I get the following table

x P(Xx)
0 0.125
1 0.375
2 0.375
3 0.125
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(C) The Distributions of Random Variables

• ex 2. Of the 15 light bulbs in a box, 5 are
  defective. Four bulbs are chosen at random from
  the box. Let the random variable, X, represent
  the number of defective bulbs selected. Construct
  a table and graph to represent this distribution.
• (NOTE the events are NOT independent (as
  selecting a defective bulb first, now influences
  the probabilities of the selection of a second
  bulb ? therefore, binompdf(4,1/3,x) will give us
  different answers than the following approach
• the number of ways of selecting x defective bulbs
  from the 5 is C(5,x)
• the number of ways of selecting (4 - x)
  non-defective bulbs is C(10, 4-x)
• the number of ways of selecting 4 bulbs from 15
  is C(15,4)
• so our basic probability formula would be ( of
  specific events) ) (total of events) C(5,x)
  x C(10,4-x) ) C(15,4)

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(C) The Distributions of Random Variables

• I get the following table

• I get the following graph

x P(X x)
0 0.154
1 0.440
2 0.330
3 0.073
4 0.004
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(D) Homework

• SL Math text, Chap 29B, p712, Q1-7
• HL Math text, Chap 30B, p730, Q1-11