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Continuous random variables

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					    Population distribution VS Sampling distribution
• The population distribution of a variable is the distribution of
  its values for all members of the population. The population
  distribution is also the probability distribution of the variable
  when we choose one individual from the population at random.

• A statistic from a random sample or randomized experiment is a
  random variable. The probability distribution of the statistic is its
  sampling distribution.

• The statistics that we will discussed the most are, the sample
                                    ˆ
  mean X , the sample proportion p and the sample variance s2.

                                  week8                               1
                  The binomial distribution
• The binomial setting:
    There is a fixed number, n, of observations.
    The n observations are independent.
    Each observation falls into one of just two categories
     (“success” and “failures”).
    The probability of a success (call it p) is the same for each
     observation.
    The binomial r.v, X counts the number of successes in n
     trials.
      Notation: X ~ Bin(n,p).
• Example:
  A biased coin (P(H) = p = 0.6) ) is tossed 5 times. Let X be the
  number of H‟s. Find P(X = 2). This X is a binomial r. v.

                                week8                                2
       Sampling distribution of a count

• When the population is much larger than the sample (at least
  20 times larger), the count X of successes in a SRS of size n
  has approximately the Bin(n, p) distribution where p is the
  population proportion of successes.

• Example 5.7 on page 317 in IPS.




                              week8                               3
    Probability function of the binomial dist.
• If X has a Bin(n, p) distribution, the probability function of X is
  given by
                         n x
           P X  x     p 1  p 
                                       n x
                                          for x = 0,1,2,…,n
                         x
• The Mean and Variance of X are,
   μX = n·p , and σX = n·p·(1-p)

• Example: The mean number of H‟s in the example above is
  μX = 5·0.6 = 3 , and the variance is σ2X = 5·0.6·0.4 = 1.2



                                week8                              4
                          Example
   You are planning a sample survey of small businesses in your
   area. You will choose a SRS of businesses listed in the
   telephone book's Yellow Pages. Experience shows that only
   about half the businesses you contact will respond.

(a) If you contact 150 businesses, it is reasonable to use the
    Bin(150, 0.5) distribution for the number of businesses X
    who respond. Explain why.
(b) What is the expected number (the mean) of businesses who
    will respond and what is its std dev.?



                               week8                             5
                          Exercise
• The probability that a certain machine will produce a defective
  item is 1/4. If a random sample of 6 items is taken from the
  output of this machine, what is the probability that there will
  be 5 or more defectives in the sample? What is the expected
  value of defective items in a sample of size 12.




                              week8                             6
                  Sample Proportions
                                                 ˆ
• The sample proportion of successes, denoted by p , is
               X
            p
            ˆ
               n
• Mean and standard deviation of the sample proportion of
  successes in a SRS of size n are
           pˆ  p               p 1  p 
                        p 
                          ˆ
                                     n

• Example 5.12 on page 322 in IPS.



                             week8                          7
         Question 1 Summer 2000, QIII b
• Suppose that the „true‟ odds are 6 to 4 that team A will win an
  upcoming Stanley Cup playoff series (so that probability of A
  winning is 0.6). You place a bet in the amount of $100 on
  team A, The payoff you will receive if team A wins is $160.
  What is your expected net gain using the quoted odds above.



• If the casino accepts 1000 bets just like yours, what is the
  expected income for the casino and the standard dev. of this
  income.



                              week8                                 8
           Question 1 Summer 2000, Q D
• While in the casino in your hotel, you try the “double till I
  win” strategy for betting. Assume that the chances are 0.5 that
  you win or lose every time you play some casino game. You
  bet $10 to start. If you win, you quit. If you lose, you double
  your bet to $20. If you win, you quit. If you lose, you double
  your bet. You quit the moment you win a game, or you will
  quit when you lose 5 consecutive times. Write down all
  possible outcomes for your evening and their probabilities.
  Workout your net gain for each outcome above.
  What is your expected net gain.




                              week8                                 9
                         Exercise
 A golf ball manufacturer is considering whether or not he
should change to a new production process. Eight percent of
the balls produced by the old process are defective and cannot
be sold while in the new process it is only five percent. But the
cost of production in the new process is 90 cents per ball while
in the old process it is 60 cents. The balls are sold at $2.00
each.
If the manufacturer wishes to maximize his expected profit,
which process should he use?




                             week8                             10
                        Exercise
A set of 10 cards consists of 5 red cards and 5 black cards. The
cards are shuffled thoroughly and I am given the first four
cards. I count the number of red cards X in these 4 cards.
The r. v. X has which of the following probability
distributions?

a) B(10, 0.5)
b) B(4, 0.5)
c) None of the above.




                            week8                              11
                           Exercise
• There are 20 multiple-choice questions on an exam, each
  having responses a, b, c, and d. Each question is worth 5
  points. And only one response per question is correct. Suppose
  that a student guesses the answer to question and her guesses
  from question to question are independent. If the student needs
  at least 40 points to pass the test. What is the probability that
  the student will pass the test?


• What is the expected (mean) score for this student.




                               week8                             12
 Normal approximation for counts and proportions
• Draw a SRS of size n from a large population having
  population p of success. Let X be the count of success in the
  sample and p  X n the sample proportion of successes. When
                ˆ
  n is large, the sampling distributions of these statistics are
  approximately normal:
                    
   X is approx. N np,     np1  p     
                         p1  p  
   p is approx. N  p,
   ˆ                                
                            n      
                                   
• As a rule of thumb, we will use this approximation for values
  of n and p that satisfy np ≥ 10 and n(1-p) ≥ 10 .

                              week8                               13
                         Example
• You are planning a sample survey of small businesses in your
   area. You will choose a SRS of businesses listed in the
   telephone book's Yellow Pages. Experience shows that only
   about half the businesses you contact will respond.
(a) If you contact 150 businesses, it is reasonable to use the
   Bin(150; 0.5) distribution for the number X who respond.
   Explain why.
(b) What is the expected number (the mean) who will respond?
(c) What is the probability that 70 or fewer will respond?
(d) How large a sample must you take to increase the mean
    number of respondents to 100?




                             week8                           14
                           Exercise
    According to government data, 21% of American children
   under the age of six live in households with incomes less than
   the official poverty level. A study of learning in early
   childhood chooses a SRS of 300 children.
(a) What is the mean number of children in the sample who come
   from poverty-level households? What is the standard deviation
   of this number?
(b) Use the normal approximation to calculate the probability that
   at least 80 of the children in the sample live in poverty. Be
   sure to check that you can safely use the approximation.




                               week8                            15

				
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