Hints for Homework Assignment 9

General Comments

• Type I: Given a description of definition of a continuous r.v., find its density or c.d.f. To do this, first determine the range of the r.v., compute the c.d.f., via the definition F(x)=P(X<=x) for x in the range, then take the derivative to obtain the density function. Note that for continous r.v., the density function cannot (usually) be computed directly. Always compute the c.d.f. first, even if the problem didn't ask for it.
• Type II: Given the density (or c.d.f.) of a continuous r.v., find a probability involving this r.v., or compute expectations, variances, etc. This requires using the appropriate formulas which express the quantities sought as integrals involving the density function. To do this correctly, you obviously need to know how to do integrals, including improper integrals. Also, it is very important here to take into account the range of the density function. If the density is defined by two different formulas for different ranges, then you need to split the integral accordingly. The same applies if the formula for the density involves |x|; in that case, consider the ranges x>0 and x<0 separately.

Specific Comments

• Problem 1: Here are some hints: (i) Heed the above advice of computing first the c.d.f., and then the density. (This applies to both (a) and (b)!) (ii) To get an idea of how to handle Y, sketch the range of values u in [0,1] for which the minimum of u and 1 - u is less than a prescribed number, say, 1/4, and find the probability that u falls into this range.
• Problem 2: This is a two stage problem. The first stage is a simple problem involving the normal distribution with unknown parameters, mathematically identical to one done in class (involving weights of a population). The second stage is a simple S/F problem that does not require any approximation.
• Problem 3: A routine problem, similar to one I did in class.
• Problem 4: Part (a) is an easy problem involving the exponential distribution, similar to the radioactive decay problem worked out in class; part (b) is also similar to a problem worked out in class (on 4/23).
• Problem 5: Part (a) is a routine application of the formulas for the expectation of a continuous r.v., and of a function of a continous r.v., similar to the last example done in Monday's class; part (b) is of the same type as the computation of the density of a log-normal distribution (worked out in Monday's class).
• Problem 6: Think of the road as the interval [0,100], with the ambulance station located at the point x=30. Note that the response time is directly correlated to the distance from the ambulance station. Therefore the probabilities asked for in (a) and (b) can be expressed as probabilities that the accident happens in certain subintervals of [0,100], and by the uniform distribution, those probabilities are proportional to the lengths of those subintervals. Parts (c) and (d) are routine, once you know the c.d.f. function of T (and the latter can be obtained from (b)).
• Problem 7 (Bonus problem): No hints given here (otherwise it wouldn't qualify as a bonus problem!).

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