Solutions to Homework # 2 1. a) Let us consider a sequence of n 4 outcomes of tosses of a coin. The sequence may start with T , in which case there are no three heads in a row if and only if there are no three heads in a row in the subsequence of the last
Solutions to Homework # 3 1. We are in the situation of the asymmetric random walk with absorbing barriers. When the bets are $1, the barriers are at x = 0 and x = n and the walk starts at x = k. The probability that the walk hits x = n before hitting x =
Homework # 3, Due Tuesday, January 31 1. Problem. At each turn, a gambler bets a certain amount, wins it with probability p and loses it with probability q = 1 - p. When he begins playing, he has $k for some integer k > 0 and his goal is to reach $n for s
Solutions to quiz # 4 (February 2) 1. Let F be the distribution function of X, so F (a) = P(X a). Hence 0 if a < -1 if - 1 a < 1 if a 1.
F (a) =
The graph of F is shown below.
1/3 -1 0 1 a
2. Let us introduce random variables X1 , . . . , X10 , wh
Solutions to quiz # 3 (January 26) 1. We are in the situation of the asymmetric random walk with the absorbing barriers at x = 0 and x = 3. The particle starts at k = 1, moves one step to the right with probability p = 3/5 and one step to the left with pr
Solutions to quiz # 2 (January 19) 1. Let I, II and III be the events that we choose the first, second and the third box respectively, and let R be the event that we choose a red marble. Then P(III|R) = Now, P(III R) = P(III) P(R|III) = and P(R) = P(R|I)P
Solutions to quiz # 1 (January 12) 1. 10! ways and 3!7! 7 7! then we can choose a committee of 4 from the group of remaining 7 people in = 4 4!3! ways. Altogether, there are a) We can choose a committee of 3 from a group of 10 people in = 10! 7! 10! = = 4
Solutions to Homework # 1 1. First, we note that there are 52 ways to select a poker hand of 5 cards. 5 a) There are 4 ways to choose a suit and then there are 13 ways to choose a hand of 5 this suit. Hence the probability to get a flush is 4 52 13 / 5 5
Homework # 4, Due Tuesday, February 7 1. Problem. Let F and G be distribution functions of random variables. Is it true that the functions 1 1 F + G and FG 2 2 are also distribution functions of some random variables? Please explain. 2. Problem. I try to
Homework # 2, Due Tuesday, January 24 1. Problem. a) We toss a fair coin n times. Let an be the number of outcomes which do not contain a sequence . . . HHH . . . (that is, do not contain 3 or more heads in a row). Find a recurrence formula relating numbe
Homework # 1, Due Tuesday, January 17
In Problems 1-3 express your answer in terms of factorials. 1. Problem. A poker hand consists of 5 cards chosen from the standard set of 52 cards. What is the probability that a random poker hand is a) a flush? (all 5