Learning Materials
Features
Discover
Chapter 7: Q.7.7I (page 358)
In Problem 7.70, suppose that the coin is tossed times. Let denote the number of heads that occur. Show that
Hint: Make use of the fact that,
When and are positive integers.
It has been shown that
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the pairs of people is, independently, a pair of friends with probability p, find the expected number of occupied tables.
Hint: Let equal or , depending on whether theth arrival sits at a previously unoccupied table.
A population is made up of disjoint subgroups. Let denote the proportion of the population that is in subgroup . If the average weight of the members of subgroup is , what is the average weight of the members of the population?
A deck of n cards numbered 1 through n is thoroughly shuf铿俥d so that all possible n! orderings can be assumed to be equally likely. Suppose you are to make n guesses sequentially, where the ith one is a guess of the card in position i. Let N denote the number of correct guesses.
(a) If you are not given any information about your earlier guesses, show that for any strategy, E[N]=1.
(b) Suppose that after each guess you are shown the card that was in the position in question. What do you think is the best strategy? Show that under this strategy
(c) Supposethatyouaretoldaftereachguesswhetheryou are right or wrong. In this case, it can be shown that the strategy that maximizes E[N] is one that keeps on guessing the same card until you are told you are correct and then changes to a new card. For this strategy, show that
Hint: For all parts, express N as the sum of indicator (that is, Bernoulli) random variables.
Let be the number of and the number of that occur in rolls of a fair die. Compute .
An urn has black balls. At each stage, a black ball is removed and a new ball that is black with probability and white with probability is put in its place. Find the expected number of stages needed until there are no more black balls in the urn. note: The preceding has possible applications to understanding the AIDS disease. Part of the body鈥檚 immune system consists of a certain class of cells known as T-cells. There are types of T-cells, called CD4 and CD8. Now, while the total number of T-cells in AIDS sufferers is (at least in the early stages of the disease) the same as that in healthy individuals, it has recently been discovered that the mix of CD4 and CD8 T-cells is different. Roughly 60 percent of the T-cells of a healthy person are of the CD4 type, whereas the percentage of the T-cells that are of CD4 type appears to decrease continually in AIDS sufferers. A recent model proposes that the HIV virus (the virus that causes AIDS) attacks CD4 cells and that the body鈥檚 mechanism for replacing killed T-cells does not differentiate between whether the killed T-cell was CD4 or CD8. Instead, it just produces a new T-cell that is CD4 with probability . and CD8 with probability .. However, although this would seem to be a very efficient way of replacing killed T-cells when each one killed is equally likely to be any of the body鈥檚 T-cells (and thus has probability . of being CD4), it has dangerous consequences when facing a virus that targets only the CD4 T-cells
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine