Extra Problems From Ch. 5

1) Do Problem 5.2 on p. 228 of Ross, except change 5 to 4. Show some work to justify your answer. (Don't leave your answer in terms of C. Use the fact that the pdf integrates to 1 to first determine what the value of C must be. (Hint: Use integration by parts (or use an integral table or appropriate software).))

2) Suppose that a man aiming at a target receives 10 points if his shot is within 1 inch of the target, 5 points if it is between 1 and 3 inches of the target, 3 points if it is between 3 and 5 inches of the target, and 0 points if it is farther than 5 inches of the target. Find the expected number of points scored if the distance from the shot to the target, in inches, is uniformly distributed between 0 and 10. (Hint: Don't make this complicated. First use the uniform distribution to determine the points distribution, which is discrete. Then use a simple discrete distribution result to get the desired expected value.)

3) Suppose that the number of years a radio functions properly has an exponential distribution having a mean of 8. If someone buys the radio used, and it is still functioning at the time it is bought, what is the probability that it will continue to function for at least 8 more years? Provide justification for your answer.

4) Do part (b) of Problem 5.31 on p. 230 of Ross. Provide work to justify your answer.

5) Do Problem 5.39 on p. 231 of Ross. Provide some work to justify your answer.

6) Do Theoretical Exercise 5.1 on p. 231 of Ross. Give your answer in terms of b (and don't plug in m/2kT for b). (Hints: One way to go about it is to do a calculus change of variable of integration in such a way as when you pull constants out in front of the integral sign, the remaining integral is a gamma function integral of the form given near the top of p. 218 of Ross, which can be evaluated using the result given by (6.1) on p. 218 and also the result given in Theoretical Exercise 5.21 on p. 232 of Ross (which you're free to use w/o providing justification for). Alternatively, because the integrand (when you integrate the pdf given for Theoretical Exercise 5.1) is even, you can use the fact that the integral of the integrand from 0 to infinity is 1/2 times the integral of the integrand from minus infinity to infinity, and after you insert some constants in the integrand (that you cancel out with constants put in front of the integral sign) you can view this integral (from minus infinity to infinity) as the variance of a normal random variable (that is, you can evaluate this integral by viewing it as x2 times a normal distribution pdf). Yet another approach is to do the integration using integration by parts. Some of you may find this to be the most straightforward approach.)