Transcript Lecture 12 - Statistics

Today
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Today: Exam review
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Thursday:
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Mid-Term!
1 hour and 20 minutes
Materials: pen/pencils, erasers, ruler, calculator
Example
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Brain cancer is a rare disease…about 3.1 cases per 100,000 people
per year
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Suppose a small insurance company has 150,000 clients on their
books
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What is the distribution of the the random variable denoting the
number of people getting the disease in a given year?
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How many claims should the company expect to see in a given year
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What is the probability of getting 2 or more cases in a given year?
Example
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At a company’s Christmas social function, attended by the sister of a colleague, 10
bottles of champagne were raffled off
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There were 50 people at the function
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Each of the 50 names were placed in a box and the box was shaken…the name of the
winner was drawn
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The name of the winner was drawn (they won 1 bottle) and the name was returned to
the box
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Our friend’s sister one 3 of the 10 bottles
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What is the distribution of the number of bottles won by the woman?
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Was the game fair?
Example
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Suppose instead that each of the 50 names were placed in a box 10
times (i.e., there are 50 sheets of paper to be drawn)
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The name of the winner was drawn (they won 1 bottle) and the name
was NOT returned to the box
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Our friend’s sister one 3 of the 10 bottles
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What is the distribution of the number of bottles won by the woman?
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Was the game fair?
Example
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A Time magazine article (March, 1989) sounded alarm bells about
costs in the fast-growing in-vitro fertilization industry
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At the time, the program cost $7,000 per attempt, with an average
success rate per attempt of 1 in 10
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Suppose a couple has $28,000 and will try the program until they are
successful
Example
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Write a probability function for the number of attempts made
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Compute the expected cost to the couple
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What is the probability of being childless after 4 trials?
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Suppose the couple has infinite funds available, what is the
distribution of the random variable denoting the attempt where the
program finally works?
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Find the expected cost in this situation
Example
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In an article entitled ``You Aren't Paranoid if You Think Someone
Eyes Your Every Move'' (1985), the Wall Street Journal noted that
big business collects detailed statistics on your behavior
Jockey International knows how many under-shorts people own;
Frito-Lay, Inc. knows which you eat first--the broken pretzels in a
pack or the whole ones; and, to get even more specific, Coca-Cola
knows people put 3.2 ice cubes in a glass.
Have you ever put 3.2 ice cubes in a glass? What did the Wall Street
Journal article mean by that statement?
Example
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At a small airport, customers arrive at a rate of 10 per 8 hour
working day
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Find the probability of getting 3 customers in a working day
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Find the probability of getting 8 or more in a working day
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What is the expected number of customers in a given day
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What is the probability that 3 customers arrive in a one minute
interval
Example
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In the old days, M&M’s came in 5 colors: brown, orange, yellow, tan
and green
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The manufacturer claimed the proportions were .4, .2, .2, .1, .1,
respectively
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In a random sample of 20 M&M’s, find the probability of getting 4
of each color
Example
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A carton of 12 eggs includes 2 rotten eggs
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2 eggs are selected
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Let X1 and X2 be the random variables denoting the “goodness”
(0=bad, 1=good) of the eggs in order of selection
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Find the probability function for (X1, X2)
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Are these two variables independent?
Example
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The cdf of the R.V. Y is F(y)=y/2, for 0<y<2
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Find F(3)
Find P(1/2<Y<3/2)
Find P(Y<1/2|Y<1)
Find the distribution function for Z=3Y-4