{"id":13784,"date":"2021-11-05T07:31:19","date_gmt":"2021-11-05T07:31:19","guid":{"rendered":"https:\/\/assignmenttutors.net\/blog\/bh-4-7-a-certain-small-town-whose-population-consists-of-100\/"},"modified":"2021-11-05T07:31:19","modified_gmt":"2021-11-05T07:31:19","slug":"bh-4-7-a-certain-small-town-whose-population-consists-of-100","status":"publish","type":"post","link":"https:\/\/assignmenttutors.net\/blog\/bh-4-7-a-certain-small-town-whose-population-consists-of-100\/","title":{"rendered":"(bh 4.7) a certain small town, whose population consists of 100"},"content":{"rendered":"
\n

Show your work and explain your steps.<\/p>\n

1. (BH 4.7) A certain small town, whose population consists of 100 families, has 30<\/p>\n

families with 1 child, 50 families with 2 children, and 20 families with 3 children.<\/p>\n

The birth rank of one of these children is 1 if the child is the firstborn, 2 if the child<\/p>\n

is the secondborn, and 3 if the child is the thirdborn.<\/p>\n

(a) A random family is chosen (with equal probabilities), and then a random child<\/p>\n

within that family is chosen (with equal probabilities). Find the PMF, mean, and<\/p>\n

variance of the child\u2019s birth rank.<\/p>\n

(b) A random child is chosen in the town (with equal probabilities). Find the PMF,<\/p>\n

mean, and variance of the child\u2019s birth rank.<\/p>\n

2. (BH 4.8) A certain country has four regions: North, East, South, and West.<\/p>\n

The populations of these regions are 3 million, 4 million, 5 million, and 8 million,<\/p>\n

respectively. There are 4 cities in the North, 3 in the East, 2 in the South, and there<\/p>\n

is only 1 city in the West. Each person in the country lives in exactly one of these<\/p>\n

cities.<\/p>\n

(a) What is the average size of a city in the country? (This is the arithmetic mean<\/p>\n

of the populations of the cities, and is also the expected value of the population of a<\/p>\n

city chosen uniformly at random.)<\/p>\n

Hint: Give the cities names (labels).<\/p>\n

(b) Show that without further information it is impossible to find the variance of the<\/p>\n

population of a city chosen uniformly at random. That is, the variance depends on<\/p>\n

how the people within each region are allocated between the cities in that region.<\/p>\n

(c) A region of the country is chosen uniformly at random, and then a city within<\/p>\n

that region is chosen uniformly at random. What is the expected population size of<\/p>\n

this randomly chosen city?<\/p>\n

Hint: First find the selection probability for each city.<\/p>\n

(d) Explain intuitively why the answer to (c) is larger than the answer to (a).<\/p>\n

1<\/p>\n

3. (BH 4.19) Let X \u223c Bin(100, 0.9). For each of the following parts, construct an<\/p>\n

example showing that it is possible, or explain clearly why it is impossible. In this<\/p>\n

problem, Y is a random variable on the same probability space as X; note that X<\/p>\n

and Y are not necessarily independent.<\/p>\n

(a) Is it possible to have Y \u223c Pois(0.01) with P(X \u2265 Y ) = 1?<\/p>\n

(b) Is it possible to have Y \u223c Bin(100, 0.5) with P(X \u2265 Y ) = 1?<\/p>\n

(c) Is it possible to have Y \u223c Bin(100, 0.5) with P(X \u2264 Y ) = 1?<\/p>\n

4. (BH 4.25) Nick and Penny are independently performing independent Bernoulli<\/p>\n

trials. For concreteness, assume that Nick is flipping a nickel with probability p1 of<\/p>\n

Heads and Penny is flipping a penny with probability p2 of Heads. Let X1, X2, . . .<\/p>\n

be Nick\u2019s results and Y1, Y2, . . . be Penny\u2019s results, with Xi \u223c Bern(p1) and Yj \u223c<\/p>\n

Bern(p2).<\/p>\n

(a) Find the distribution and expected value of the first time at which they are<\/p>\n

simultaneously successful, i.e., the smallest n such that Xn = Yn = 1.<\/p>\n

Hint: Define a new sequence of Bernoulli trials and use the story of the Geometric.<\/p>\n

(b) Find the expected time until at least one has a success (including the success).<\/p>\n

Hint: Define a new sequence of Bernoulli trials and use the story of the Geometric.<\/p>\n

(c) For p1 = p2, find the probability that their first successes are simultaneous, and<\/p>\n

use this to find the probability that Nick\u2019s first success precedes Penny\u2019s.<\/p>\n

5. (BH 4.34) Each of n \u2265 2 people puts his or her name on a slip of paper (no<\/p>\n

two have the same name). The slips of paper are shuffled in a hat, and then each<\/p>\n

person draws one (uniformly at random at each stage, without replacement). Find<\/p>\n

the average number of people who draw their own names.<\/p>\n

6. (BH 4.36) In a sequence of n independent fair coin tosses, what is the expected<\/p>\n

number of occurrences of the pattern HT H (consecutively)? Note that overlap is<\/p>\n

allowed, e.g., HT HT H contains two overlapping occurrences of the pattern.<\/p>\n

7. (BH 4.43) You are being tested for psychic powers. Suppose that you do not<\/p>\n

have psychic powers. A standard deck of cards is shuffled, and the cards are dealt<\/p>\n

face down one by one. Just after each card is dealt, you name any card (as your<\/p>\n

prediction). Let X be the number of cards you predict correctly. (See Diaconis<\/p>\n

(1978) for much more about the statistics of testing for psychic powers.)<\/p>\n

(a) Suppose that you get no feedback about your predictions. Show that no matter<\/p>\n

what strategy you follow, the expected value of X stays the same; find this value.<\/p>\n

2<\/p>\n

(On the other hand, the variance may be very different for different strategies. For<\/p>\n

example, saying \u201cAce of Spades\u201d every time gives variance 0.)<\/p>\n

Hint: Indicator r.v.s.<\/p>\n

(b) Now suppose that you get partial feedback: after each prediction, you are told<\/p>\n

immediately whether or not it is right (but without the card being revealed). Suppose<\/p>\n

you use the following strategy: keep saying a specific card\u2019s name (e.g., \u201cAce of<\/p>\n

Spades\u201d) until you hear that you are correct. Then keep saying a different card\u2019s<\/p>\n

name (e.g., \u201cTwo of Spades\u201d) until you hear that you are correct (if ever). Continue<\/p>\n

in this way, naming the same card over and over again until you are correct and then<\/p>\n

switching to a new card, until the deck runs out. Find the expected value of X, and<\/p>\n

show that it is very close to e \u2212 1.<\/p>\n

Hint: Indicator r.v.s.<\/p>\n

(c) Now suppose that you get complete feedback: just after each prediction, the card<\/p>\n

is revealed. Call a strategy \u201cstupid\u201d if it allows, e.g., saying \u201cAce of Spades\u201d as a<\/p>\n

guess after the Ace of Spades has already been revealed. Show that any non-stupid<\/p>\n

strategy gives the same expected value for X; find this value.<\/p>\n

Hint: Indicator r.v.s.<\/p>\n

8. (BH 4.66) Use Poisson approximations to investigate the following types of coincidences.<\/p>\n

The usual assumptions of the birthday problem apply, such as that there<\/p>\n

are 365 days in a year, with all days equally likely.<\/p>\n

(a) How many people are needed to have a 50% chance that at least one of them has<\/p>\n

the same birthday as you?<\/p>\n

(b) How many people are needed to have a 50% chance that there are two people<\/p>\n

who not only were born on the same day, but also were born at the same hour (e.g.,<\/p>\n

two people born between 2 pm and 3 pm are considered to have been born at the<\/p>\n

same hour).<\/p>\n

(c) Considering that only 1\/24 of pairs of people born on the same day were born<\/p>\n

at the same hour, why isn\u2019t the answer to (b) approximately 24 \u00b7 23? Explain this<\/p>\n

intuitively, and give a simple approximation for the factor by which the number of<\/p>\n

people needed to obtain probability p of a birthday match needs to be scaled up to<\/p>\n

obtain probability p of a birthday-birthhour match.<\/p>\n

(d) With 100 people, there is a 64% chance that there are 3 with the same birthday<\/p>\n

(according to R, using pbirthday(100,classes=365,coincident=3) to compute<\/p>\n

it). Provide two different Poisson approximations for this value, one based on creating<\/p>\n

an indicator r.v. for each triplet of people, and the other based on creating an<\/p>\n

indicator r.v. for each day of the year. Which is more accurate?<\/p>\n

3<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Show your work and explain your steps. 1. (BH 4.7) A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these children is 1 if the child is the firstborn, 2 if […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31],"tags":[],"class_list":["post-13784","post","type-post","status-publish","format-standard","hentry","category-homework-help"],"_links":{"self":[{"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/posts\/13784","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/comments?post=13784"}],"version-history":[{"count":0,"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/posts\/13784\/revisions"}],"wp:attachment":[{"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/media?parent=13784"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/categories?post=13784"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/assignmenttutors.net\/blog\/wp-json\/wp\/v2\/tags?post=13784"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}