Hints for Homework Assignment 6

General Remarks

Relevant reading: Before working on the assignment, review your class notes, and read Section 3.1, p. 139 - 141, 144 - 147, including the examples in those sections.

Finding distributions: If a problem asks for a distribution of a r.v. (i.e., is a "type I" problem), the first (easy) step should always be to write down the list of values of this r.v. The second (and much harder) step is to compute, for each value on this list, the probability that the r.v. takes on this value. In most cases, it is not a good idea to try to do this computation "wholesale", using a general x. It is usually better to take the values one at a time, and perform the corresponding probability computation. After a few such computations, you may see a pattern; if the pattern is obvious, it is okay to skip the details for the remaining probability computations, and just write down the general formula. Similarly, to find a joint distribution of X and Y, you have to write down a list of all values x of X, and all values y of Y, and then, for each such pair (x,y), compute the associated probability P(x,y).

Representing distributions: A distribution (or joint dist.) can be represented by a distribution table (or matrix in case of a joint dist.), or by a formula, which may involve several cases, along with a range of values. The range is essential if one wants to use the distribution for probability computations, and a formula alone, without a range, does not adequately define a distribution. Thus, be sure to include the range for any distribution you compute; points will be taken off if the range is missing. A joint distribution can be represented in matrix form, but if there are more than two or three values involved this becomes rather unwieldy. In those cases, it's better to represent a joint distribution by a formula for P(x,y) (possibly involving several cases), such as "P(x,y)=1/100 for x=1,2,...,100, and y=x+1 or x-1, and P(x,y)=0 otherwise". Be sure to include the range for x and y.

Checking distributions: If you have to compute a distribution (ordinary or joint), check (if possible) whether the probabilities add up to 1. If they don't, you made a mistake.

Specific Comments

• Problem 1: Part (a) is a very simple. In part (b), max(a,b) means the larger of the two numbers a and b. For example, max(2,3)=3, max(2,2)=2. Similarly, min(a,b) means the smaller of the numbers a and b. Part (b) is essentially a simplified version of Example 4 in 3.1. To check your work, add up the probabilities; the sum should be 1.
• Problem 2: Note that the problem involves 3 tosses, not 4. For example, if the outcome of the three tosses is the sequence HHT, then X=2, and Y=1. Part (a) reduces to a very simple "assembly level" computation within a S/F model. To check your answer to (c), add up the probabilities in the distribution you have calculated and see if the sum is 1.
• Problem 3: This reduces to an ordinary probability computation. As mentioned earlier, you should not try to get a general formula for P(X=x), but compute separately P(X=2), P(X=3), etc. These probability computations are somewhat similar to those in the "number guessing" problem in HW 5. The key is to see what (for example) X=4 means in terms of the sequence of tickets drawn. You can think of the tickets as marked 1a, 1b, 2a, 2b, etc. When you are done with the computations, check if the probabilities add up to 1.
• Problem 4: You should do part (a) first, and then solve (b), using the result of (a). This problem is a bit unusual in that normally one would first find the distribution of X, and then use that distribution to compute probabilities such as (1) P(X>k), whereas here it is the other way around. The reason for this is that probabilities of type (1) are much easier to compute than the probabilities (2) P(X=k) arising in the distribution. For part (b), you have to express (2) in terms of probabilities of type (1); this turns out to be quite easy. Part (a) is trickier. The key is to properly interpret and rephrase the event X>k. As always, it's a good idea to first work the problem for a concrete value of k, say k=5. The situation then is quite similar to that of the birthday problem variant in a previous HW assignment, which asked for the probability that it takes more than 5 people to find a matching birthday.
• Problem 5: Part (a) is a slight generalization of the world series problem that came up in an earlier HW assignment. Part (b) is easy, using probabilities computed in (a). Part (e) boils down to 4 separate probability computations, each of the type encountered in part (a). (Skip the "distribution check" for the last part.)
• Problem 6: Part (i) is a routine type II problem. Part (ii) is slightly more complicated. Part (iii) is similar to an example done in class, but involves considerably more work, and a good deal of care. (Hint: For the values k of X+Y, distinguish between the cases k<=n+1 and k>n+1.) It may help to first consider a particular value of n, say n=3, see what is going on in this case, and then move on to the general case. For parts (ii) and (ii), the "distribution check" can be used, but it does involve some serious calculations and require a formula for summations over k from 2 to n. (You can, however, do this check in a concrete case, say n=3.)
  

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