Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils,...
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Transcript of Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils,...
![Page 1: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/1.jpg)
Today
• Today: Exam review
• Thursday: – Mid-Term!
– 1 hour and 20 minutes
– Materials: pen/pencils, erasers, ruler, calculator
![Page 2: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/2.jpg)
Example
• Brain cancer is a rare disease…about 3.1 cases per 100,000 people per year
• Suppose a small insurance company has 150,000 clients on their books
• What is the distribution of the the random variable denoting the number of people getting the disease in a given year?
• How many claims should the company expect to see in a given year
• What is the probability of getting 2 or more cases in a given year?
![Page 3: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/3.jpg)
Example
• At a company’s Christmas social function, attended by the sister of a colleague, 10 bottles of champagne were raffled off
• There were 50 people at the function
• Each of the 50 names were placed in a box and the box was shaken…the name of the winner was drawn
• The name of the winner was drawn (they won 1 bottle) and the name was returned to the box
• Our friend’s sister one 3 of the 10 bottles
• What is the distribution of the number of bottles won by the woman?
• Was the game fair?
![Page 4: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/4.jpg)
Example
• 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)
• The name of the winner was drawn (they won 1 bottle) and the name was NOT returned to the box
• Our friend’s sister one 3 of the 10 bottles
• What is the distribution of the number of bottles won by the woman?
• Was the game fair?
![Page 5: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/5.jpg)
Example
• A Time magazine article (March, 1989) sounded alarm bells about costs in the fast-growing in-vitro fertilization industry
• At the time, the program cost $7,000 per attempt, with an average success rate per attempt of 1 in 10
• Suppose a couple has $28,000 and will try the program until they are successful
![Page 6: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/6.jpg)
Example
• Write a probability function for the number of attempts made
• Compute the expected cost to the couple
• What is the probability of being childless after 4 trials?
• Suppose the couple has infinite funds available, what is the distribution of the random variable denoting the attempt where the program finally works?
• Find the expected cost in this situation
![Page 7: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/7.jpg)
Example
• 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?
![Page 8: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/8.jpg)
Example
• At a small airport, customers arrive at a rate of 10 per 8 hour working day
• Find the probability of getting 3 customers in a working day
• Find the probability of getting 8 or more in a working day
• What is the expected number of customers in a given day
• What is the probability that 3 customers arrive in a one minute interval
![Page 9: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/9.jpg)
Example
• In the old days, M&M’s came in 5 colors: brown, orange, yellow, tan and green
• The manufacturer claimed the proportions were .4, .2, .2, .1, .1, respectively
• In a random sample of 20 M&M’s, find the probability of getting 4 of each color
![Page 10: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/10.jpg)
Example
• A carton of 12 eggs includes 2 rotten eggs
• 2 eggs are selected
• Let X1 and X2 be the random variables denoting the “goodness” (0=bad, 1=good) of the eggs in order of selection
• Find the probability function for (X1, X2)
• Are these two variables independent?
![Page 11: Today Today: Exam review Thursday: –Mid-Term! –1 hour and 20 minutes –Materials: pen/pencils, erasers, ruler, calculator.](https://reader038.fdocuments.us/reader038/viewer/2022110401/56649de45503460f94adabce/html5/thumbnails/11.jpg)
Example
• The cdf of the R.V. Y is F(y)=y/2, for 0<y<2
– 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