Suppose that there are 12 types of coupons and that each time one obtains a coupon, it is, indepe...
Suppose there are 25 different types of coupons and suppose that each time one obtains a coupon, it is equally likely to be any of the 25 types. Compute the expected number of different types that are contained in a set of 10 coupons.
4. Suppose you continually collect coupons and that there are two different types of coupon, type A and type B. Suppose also that each time a new coupon is obtained it is a type A coupon with probability 1/3 and a type B coupon with probability 2/3, independently of what coupons you have collected so far. Let X be the number of coupons collected until you have at least one coupon of both types. (a) Find the probability mass function...
Suppose there are n type of coupons. Each new coupon collected is of type i with probability Pi, independently of any other collected coupon. Here, D=1 Pi = 1. Suppose k coupons are collected. Let A be the event that there is at least one coupon of type i among the k collected. For i #j, (1) Compute P(A|AU A;) (2) Compute P(A|Aj)
1. Suppose there are m 2 1 different types of coupons, and a total of n coupons is to be collected. Each new coupon collected is, independent of the past, a type i coupon with probability i, 1 < i< m. Define for i-1,... ,m, Х,-{ 0, otherwise 1, if at least one type i coupon is among the n collected, type Let X = Xut + x,n. Calculate E(X) and Var(X).
Problem 1. Stanislaw is collecting coupons. Each day he receives randomly one of n distinct coupons with equal probabilities (independently of other days (a) Let T be the number of days it takes Stanislaw to obtain a complete set. Explain why T can be written as a sum of n independent Geometric random variables and say what their parameters are (b) Compute the expected value of T. (Use the fact that the expectation of a sum of random variables is...
Q. Given 25 different type of coupons, one coupon is obtained each time. one set obtain 10 coupon. The probability that the type i coupon is not in the set is 24C10/25C10. Is it right? (24C10 => combination) If wrong, please tell me the answer. (i is range 1~25. coupon type index)
Exercise 12.6 At each stage, one can either pay 1 and receive a coupon that is equally likely to be any of n types, or one can stop and receive a final reward of jr if one's current collection of coupons contains exactly j distinct types. Thus, for instance, if one stops after having previously obtained six coupons whose successive types were 2, 4, 2, 5, 4, 3, then one would have earned a net return of 4r -6. The...
The coupon collector problem calculates the expected number of days it takes to get n different coupons, if one receives one of the n coupons at random each day. The number of days is approximately n(0.577 + ln n). Use this to calculate the expected number of TCP connections a random port scan (scanning port numbers 0 through 1023) needs to eventually check all 1,024 well- known port numbers.
(12 points) A software function generates a random number N digits long. Each digit is determined by randomly selecting a value from 0 through 9. All ten values are equally likely, and different digits may have the same value. Determine the minimum length N such that there is at least a 50% probability that at least one digit will have the value 0.
1. Consider the coupon game. But suppose that instead of
decisions being made simultaneously, they are made sequentially,
with Firm 1 choosing first, and its choice observed by Firm 2
before Firm 2 makes its choice.
a. Draw a game tree representing this game.
b. Use backward induction to find the solution. (Remember that
your solution should include both firms’ strategies, and that Firm
2’s strategy should be complete!)
2. Two duopolists produce a homogeneous product, and each has a...