RMI 3750 :Problem Set 2, due 9/23/2015

1. Two jars contain coins. Jar I contains 7 pennies, 6 nickels and 8 dimes. Jar II contains 5 pennies, 3 nickels and 1 dime. A jar is selected at random and a coin is selected from that jar. If the coin is a nickel, what is the probability that it came from jar II?

2. A card is drawn from a standard 52 cards deck, not replaced, and a second card is drawn. What is the probability that the second card is a spade?

3. A probability distribution for the claim sizes for an auto insurance policy is given in the table below

Claim Size 20 30 40

50 60 70 80

Probability 0.20 0.10 0.10

0.20 0.15 0.10 0.15

a) Evaluate the cdf for this distribution at the following points: 0, 20, 30, 40, 50, 90.

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b)

c)

What is the mean of this distribution?

What is the variance of this distribution?

d) the mean claim size? (In other words, what percentage of the claims are greater than or equal to [the mean size minus one standard deviation] but less than or equal to [the mean size plus one standard deviation]?)

What percentage of the claims are within one standard deviation of

4. Define a random variable Y as the outcome of a fair die roll times 3 (e.g., if a five is rolled, Y takes a value of 5*3=15). What is the expected value of Y? What is the variance of Y?

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5. A random variable X has E(X)=3 and V(X)=7. Define a new random variable: Y=6X + 3. What is E(Y) and V(Y)?

6. In class we often talked about a 6-sided die. Now consider a 24-sided die, with sides numbered 1 to 24 (so that the sample space of outcomes is {1,2,…,24}). Suppose that all outcomes are equally likely. What is the mean and variance of the outcome of a roll of a 24-sided die?

7. A firm’s revenue distribution for the coming year is (in million dollars): 15% probability of 5, 25% probability of 10, 20% probability of 12, 30% probability of 14 and 10% probability of 20. If F(.) is the cumulative distribution function of this random variable, what is F(11) equal to? What is F(14)? What is the variance of the firm’s revenue distribution?

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8. Airtran’s flight #307 can accomodate 50 passengers, but the flight is over- booked, as 52 tickets were sold. Each ticketed passenger can arrive late and miss the flight with a probability 0.02. What is the probability that no passenger arrives late? What is the probability that exactly one passanger arrives late? What is the probability that Airtran has to pay overbooking fees (and reschedule passengers to different flights)?

9. The world incidence of (frequency of world population having) diabetes is 5%. If 20 persons are chosen at random, what is the probability that no more than 3 have the disease?

10. Plot in Excel the probability mass function (density) for the following:

• a Binomial random variable X with n = 10, p = 0.3 • a Binomial random variable Y with n = 100, p = 0.4 • a Uniform random variable X with n = 50

Attach the clearly labeled graphs.