Analysis CS504: of Computations and Systems Spring 1999
Sample problems, use for exam? March 3, 1999
1. A basketball team has 12 players. In our version of the rules a team is allowed up to 5 exchanges during the entire game, and an exchanged player may not return later. (a) How many different openning-ve sets can the team have? (b) How many different game plans can the team have of each size (the size of a game plan is the total number of players it uses 5 to 10, naturally)? Total? [A game plan species the opening ve, how many exchanges are to be done, and the identities of the players taking part in each exchange. Order, except among the openning ve, is important.] (c) How is the calculation of part (b) affected when there is one player who is so crucial to this team that no plan ever calls to exchange him... (d) What is the value of P, the probability of a player to play at all, during one game, assuming his likelihood of openning is p0 , and at each exchange the entering player is selected with equal probability from among those yet eligible? Naturally this is a function of the number of exchanges in the game plan. call this number s and write a formula for P in terms of p0 and s. (e) Now assume the manager is strong-headed and plans to use the same number of exchanges s throughout the season (all K games!). What is the probability of the player of part (d) to play at least once during the season? In at least half the games? In all the games?
2. Assume you set out to perform a numerical calculation involving binomial coefcients. (a) You plan to use for them the integer representation provided by the computer (if you were writing in C you would dene them as an array of long ints. Assume thatas is the case in many architecturesthe machine provides you with 32 bits for such a value, in twos complement representation, which means that the largest int it stores is 231 1. What is the largest n for which n k n? k can be thus represented for all 0 (b) Now you are willing to trade off precision for size and use the oating point unit to manipulate these numbers. Hence they are to be stored in the standard oating point representation. We further assume the machine always normalizes before it stores FP numbers and that it uses t bits in the exponent eld of this representation. Further, it uses the excess-exponent representation, which means that the actual exponent is 2t 2 less than the number stored in the exponent eld. Now what is the largest n, as above, given in terms of t? What are the actual values when t 8 12 24 or 32?
3. We have proved in class the Chebyshev inequality. Look at the proof and nd for which kind of random variable inequality the will be achieved as an equality. Explain.
CS 504:
Sample problems.......
2
4. Show the following sums. Try do it in two ways: directly (where summation by parts could be helpful) and by mathematical induction.
k 1
n
2 1 Hn
k 1
k 1
n
k Hk m
n
n 1 m 1
k
Hk 1 k
1 2 H 2 n
The last quantity here, Hn is the nth second-order harmonic number, and equals k n1 1 k2 . 5. Let X be a random variable with the binomial distribution B n p . You may think of it as the number of heads coming up when a coin is ipped n times, where it has the probability of p turning heads up each and every time, independently. Let un (we leave the p implicit in this notation) be the probability that X is even 0 or 2 etc. What is the value of un ? (a) Since the PMF of X is known, you can write a sum for it and compute it directly. This kind of sum is known as bisection, since it picks every other term in a sequence. (b) Another way is to write a recursion for un , by considering how it can depend on un 1 . Write the recursion, solve it, obtain the same value as in part (a).
A few problems from the text which I nd suitable...
0 and a0
7. Prob. 2.60, asks to plot a divide and conquer recurrence. Instead, solve it for values of the index which are powers of two. Note that Figs. 2.1 and 2.2 there illustrate the claim I made in class, that it is enough to solve such recurrences for such values of the index in order to track well the evolution of the sequence as it evolves. In truth, this only holds for sequences which increase monotonically (or decrease into the n...