Statistics Problems

1. If the number of minutes that a doctor spends with a patient is a random variable having an exponential distribution with the parameter θ = 9, what are the probabilities that it will take the doctor at least 20 minutes to treat (a) one patient; (b) two patients;

2. A lawyer has an unlisted number on which she receives on the averages 2.1 calls every half-hour and a listed number on which she receives on the average 10.9calls every half-hour. If it can be assumed that the numbers of calls that she receives on these phones are independent random variables having Poisson distributions, what are the probabilities that in half an hour she will receive altogether (a) 14 calls; (b) at most six call;

3. Let X be the amount of premium gasoline (in 1000 gallons) that a service station has in its tanks at the beginning of a day, and Y the amount that the service station sells during that day. If the joint density of X and Y is given byf(x,y) = 1/200, for 0<y<x<20, use the distribution function techniques to find the probability density of the amount that the service station has left in its tanks at the end of the day.

4. If X is the total we roll with a pair of dice,find the probability distribution of the remainder we get when the values of X are divided by 3. (The probability distribution is given on the text but is too long to type it here.I know that the possibleremainder we can get is 0, 1, and 2. The probability of each one should be thesame so g(0)=g(1)=g(2) =1/3. Is it right? I mean how should we explain this?)

5. If the joint probability distribution of X and Y is given by f(x,y) = (xy)/36 for x=1,2,3 and y=1,2,3, find (a) the probability distribution of XY; (b) the probability distribution of X/Y