Today

• Today: Exam review

• Thursday: – Mid-Term!

– 1 hour and 20 minutes

– Materials: pen/pencils, erasers, ruler, calculator

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?

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?

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?

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

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

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?

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

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

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?

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