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The Bayesian Way Introductory Statistics For Economists And Engineers 1st Edition Svein Olav Nyberg - Solutions
Let X be a stochastic variable taking the values −2,−1, 0, 1, 2 with probability 1/5 each, and let Y = X2. Are X and Y independent? What is ρXY ?
Find μZ and σZ, when ... (a) ZX-Y, Mx = 4, x = 3, y = -3, y = 2; (b) Z = X+X+ X3, and = 0x = 4, 0x3 = 12; MX (c) Z=1Xk and x = k and x = =1 (d) Z = X and MXk X and x = 14 and x = 4; ox (e) Z=kX and x = and x = 0; (f) Z=1X and x = k and MX2 = 2, MX3 = 3, whereas = 3, k; = k. Xk
Let X, Y be independent stochastic variables with standard deviation respectively σX and σY, and let Z = aX + (1 − a)Y. Let σZ be Z’s standard deviation. For which value(s) of a is σZ the smallest?
You have a fair \(D_{60}\) die with the numbers \(0 \ldots 59\) printed on each face. Find the expected value and standard deviation of the outcomes from a single toss, and find the probability that 4 is one of its digits.
For which value of the parameter \(p\) does a Bernoulli distributed stochastic variable \(X\) have the largest expected value \(\mu_{X}\) ?
1. What is the purpose of statistics?2. What is a D8?3. Given a D4 and a D6: in how many ways can you get a total of five?
Find the symmetrical alternatives when you flip a coin three times. What is the probability of two heads and one tails?
Find the symmetrical alternatives when you toss two D6. What is the probability that their sum is three? What is the probability their sum is seven?
If we could make pancakes forever, and it turns out that the proportion of burned pancakes stabilizes at 0.137, would that matter for the probability that pancakes are burnt?
(a) Explain in your own words what Σx, Σx2 , SSx, ̄x, and ̃x are. Why do we have different formulas for these quantities?(b) What is the difference between population and sample, and how are they connected?
Find all the three measures of location (mode, median, mean), and decide which is the most suitable one for the situation.(a) The members of Femund Fishers’ Union are located as follows:i. 24 are from Drevsjø in Engerdal, which has postal code 2443 ii. 6 are from ˚ Alesund, which has postal
For the data sets below:Calculate the median and the interquartile range.Find the mean and the sample standard deviation.(a) {−1,−3, 4}(b) {−0.2, 9.6,−0.1, 11.1, 1.3,−0.2, 11.1,−0.8, 0.4}(c) {60, 66, 70, 103, 138, 34}(d) {0.971 49, 0.659 64, 0.345 81, 0.515 90, 0.928 81}
We have written the data sets below as pairs of lists: v = {v1,…, vn} and a = {a1,…, an}, meaning a1 observations of value v1 etc.Set up a frequency table and bar chart.Set up a cumulative frequency table and a cumulative bar chart.Calculate the median and the interquartile range.Calculate the
You are in charge of a joint purchase of retro sports jackets for local FC Bayern Munich supporter club. The sizes correspond to chest measurements, and are (in cm): S=87-94,M=94-102, L=102-110, XL=110-121, XXL=121-133, 3XL=133-145. A few of the members are interested in making orders, and the
We have written the data sets below as pairs of lists, a = {a1,…, an} and I = {(l1, u1),…, (ln, un)}, meaning a1 observations in interval I1 etc.Make the cumulative table and graph.Calculate median and interquartile range.Find the mean and the sample standard deviation.Mark these measures on
A randomized survey among the supporters of the Scottish football team Heart of Midlothian FC yielded the following numbers in the different age groups: 0-12 years: 1, 13-18: 9, 19-34: 41, 35-50: 58, 51-64: 33, 65-80: 2. Use the formulas for grouped data in the following calculations:(a) Create a
This is a practical exercise, where you need a big bag of gummy worms or equivalent a measuring tool (e.g., a ruler or measuring tape) For each gummy worm, stretch the gummy worm over the measuring tool until it snaps write down the colour, and the length at which it snapped When you are done
(a) What is the difference between covariance and correlation?(b) What is “centered form”?(c) What is SSe a measure of?
You are given the data set D = {(x, y)}i∈I = {(−1, 3), (0, 5), (3, 9), (5, 7)}.(a) Set up the data in a table.(b) Plot the data in a diagram.(c) Calculate the covariance between x and y (both the population and the sample versions).(d) Calculate the correlation between x and y.
You are given the data set D = {(x, y)}i∈I = {(−1, 3), (0, 5), (3, 9), (5, 7)}.Find the linear regression line. (a) Write the data in centered form, D = {(x,y)}iel (b) Find the regression coefficient by employing Rule 3.3.5. (c) Find the regression line in centered form by employing Method
For the data sets below, calculate the following.The covariances σxy and sxy.The correlation between x and y.The linear regression line. Use matrix regression.The square of the standard error, s2e.Illustrate at least one of them with a regression line and data points.(a) D = {(85, 221.5), (103,
For the data sets below, calculate the following.The linear regression surface. Use matrix regression.The square of the standard error, s2e.(a) D = {(42, 79, 1056), (62, 51, 564), (57, 47, 507), (37, 49, 655), (17, 26, 337), (39, 78, 1155), (43, 43, 593), (20, 13, 174), (97, 52, 485), (82, 94,
(a) What is a set?(b) What is a sequence?(c) What is a combination?(d) Why does (e) What are the connections between binomial and multinomial? n n (*) = (x)? n-k
A = {a, b, c, d, e, h, i, j}, B = vowels, C = letters with an even numbered place in the alphabet. Find A ⧵ (B ∩ C).
Kari and Mona are looking at who in their class they have beaten at arm wrestling. Kari has beaten 15, wheras Mona has beaten 13. Of these, 7 have been beaten by both Kari andMona.(a) Howmany have been beaten by at least one of them?(b) Call the set of classmates beaten by Kari, K, and the set
Palle and Jens are looking at which capitals they have visited. Palle has visited 17 capitals, whereas Jens has visited 23. The number of capitals visited by at least one of them is 33.(a) How many capitals have been graced by a visit from both?(b) Call Palle’s capitals P, and Jens’ capitals J.
In the Scottish village Glenwhisky there are many pubs. Half of the pubs serve Dalwhinnie, and a third serve Laphroig. Only 5% of the pubs serve both Dalwhinnie and Laphroig.(a) Angus MacAbstainer drinks only these two brands of whisky, and is otherwise a teetotaler. What is the proportion of pubs
In your church’s stock of hymnals, 35 of the hymnals are of the old edition.Half the hymnals have a flyer about your upcoming Christmas concert.Nine out of ten of your church’s hymnals are either old or have a flyer about your Christmas concert.(a) What is the proportion of hymnals that also
A = {red, orange, green, indigo, violet}, and B = {yellow, green, blue}. Ωconsists of all seven colors of the rainbow, and is our universe. (a) What is Ac? (b) What is AUB? (c) What is An B? (d) What is A \ B? (e) What is B\A? (f) Is it true that "green A"? (g) Is it true that "yellow A"? (h) Is
Calculate 28 (2,3,5,7,11) 2,3,5,7,11-
Calculate 231 4
The jedi master N’s light saber display at a small venue at your university is fully booked, and the arrangement committee decides to expand with an extra show, and to divide the hopeful viewers into two batches. There is only room for 58 at the first show (which is also everyone’s primary
You are sports dictator in the UK for a day, and have decided that precisely 7 of the next 12 Premier League (European) football matches should be home wins. In how many ways can you pick those 7 games?
Your university has a mid-semester break (which everyone knows means a self study week). You are taking 7 subjects this semester, but knowing yourself, you decide to pursue 3 of them over the week. How many different combinations of 3 subjects can you choose, from the 7 you are following?
You have a CD collection of 60 CDs, but as you are going to your cabin you find that your bag can hold at most 13. You decide to bring the maximum number; how many different combinations of CDs can you bring?
You are playing 5-dice Yatzee. How many different full houses are possible?
You are playing strip poker. Wearing briefs only, you realize your hand is terrible. You decide to trade in 3 of your cards. Any, since you consider them all equally bad. How many different sets of 3 cards is it possible to discard from your hand of 5?
You are deer hunting with 6 buddies. As a deer shows up at the edge of the clearing, you all pull your rifles and shoot. The deer dies instantly. As you later quarter the deer, you find only 3 bullets. How many different combinations of hunters could have contributed to the kill?
In Norway, a phone number consists of 8 digits, where the first cannot be 0. A phone number was recently sold for 1 million NOK. It consisted of a number, followed by 6 equal digits different from the first, before the first digit was repeated.(a) How many such phone numbers are possible?(b) Can
Cinema: Ewan and Aiden notice that they have both bought tickets to the same movie, and both of them in row 13, which has 25 seats.(a) How many different ways can Ewan and Aiden be placed in row 13?(b) How many of these ways put them next to one another?(c) (Chapter 5)If the choice of tickets is
There are 3 cookies left on the table. You realize that if you are quick, you may get away with grabbing two before someone else grabs the third.(a) How many pairs of cookies is it possible to pick from the 3 (the order is irrelevant)?(b) What if the other guy got to grab one cookie, whereupon you
You see 8 attractive women on the beach. You know you’ll get around to inviting 3 of them for a barbecue party before your friend Bram has invited the remaining 5 to his barbecue party.(a) How many different combinations of your 3 invitations are possible, given the 8 women?(b) How many different
You have decided to divide a deck of cards into two piles; one of 22 cards, and one of 30.(a) You pick the 30 cards for pile 1, and the remaining 22 are put into pile 2. How many divisions into 2 such piles are possible?(b) You decide to instead pick 22 cards for pile 1, and let the remaining 30 be
There are 70 white roses on a rosebush. You are either painting them red, or leaving them white.(a) You are painting 47 roses red (and leaving the rest white). In how many ways can you do that?(b) You are leaving 23 white (and painting the rest red). In howmany ways can you do that?(c) Why are
The number of ways to sample s elements from a set of n possible, unordered, and without replacement, has a certain formula.(a) What is this formula?(b) What is the formula for sampling n − s elements from a set of n possible?(c) Why are the answers to these two questions the same?
You and your friends Gregory, Bear, and Arne are doing a project together.You have subdivided the project into 34 tasks, and you have decided that you’ll do 8 tasks, Gregory will do 5, Bear will do 9, and Arne will do 12.How many ways are there to divide the tasks among you in this fashion?
In how many ways can you sample 5 elements from a collection of 12, when the sampling is…(a) ordered, with replacement;(b) unordered, with replacement;(c) ordered, without replacement;(d) unordered, without replacement.(e) Give a practical example of each of the four kinds of sampling.
(a) What is an Euler diagram? What can it teach us about the basic laws of probability?(b) What is frequentist probability?(c) What is Bayesian probability?(d) What does conditional probability mean, and how does it differ from regular probability?(e) What does statistical independence mean?
Coin flipping: best done in groups, as a competition. Try to control the outcome of a coin flip, flipped from a reasonable (not too far, not too close)distance from a level surface. After a period of practice, set up a competition sequence of 10 coin flips, where you compete on who gets closest to
Resistors: pick some resistors with the same nominal resistance. Measure the actual resistance, and compare it to the nominal: is it over or under?Measure in batches of 10. Do you think you can influence the number that are over? Give reasons for why or why not. Then try it out!
The cooler: your local corner supermarket tends to be a bit negligent about checking their stock, and you know from experience that one in five milk cartons is past its sell-by date. Earlier today, you bought a carton of milk, and forgot to check the date.(a) What is the frequentist probability
The cooler II: your flat mate is going to the local corner supermarket to buy milk, and she never checks the sell-by date.(a) What is the frequentist probability that the milk she buys is past its sell-by date?(b) What is the objective Bayesian probability that the milk she buys is past its sell-by
A given coin has probability p = 0.37 of heads. What is the probability of tails?
P(A) = 0.5, P(B) = 0.25, P(AB) = 0.125.(a) What is P(A ∪ B)?(b) What is P(A|B)?(c) What is P(B|A)?
P(A) = 0.5, P(B|A) = 0.2, P(A|B) = 0.4. Find P(B).
A biased D6 die has probabilities P(1) = 0.1, P(2) = 0.1, P(4) = 0.2, P(5) = 0.2, P(6) = 0.15. What is P(3)?
(a) What is P(AB)?(b) What is P(A ∪ B)?(c) What is P(A|B)? P(A) = , P(B) = , and P(B|A) = .
P(A) = 0.3, P(B) = 0.2, and P(B|A) = 0.25.(a) What is P(AB)?(b) What is P(A ∪ B)?(c) What is P(A|B)?
P(AB) = 1/4 and P(B) = 3/4. What is P(A|B)?
P(AB) = 0.12 and P(A) = 0.6. What is P(B|A)?
Illustrate the following formulas with Euler diagrams, and give five concrete examples of each:P(A ∪ B) = P(A) + P(B) − P(AB)and P(AB) = P(A) + P(B) − P(A ∪ B).
A coin has probability p = 0.37 of heads. What is the probability of the sequence HHTHTHHHTTTHHHHTTHTHTHHT?
A coin has probability p = 0.37 of heads. What is the probability of 14 heads in 37 flips?
A coin has probability p = 0.53 of heads. What is the probability of 21 heads in 47 flips?
A coin has probability p = 0.61 of heads. What is the probability of 30 heads in 60 flips?
Assorted Candies I: you are tidying up after a party, and find a bag of assorted candies. The only pieces left in the bag are 7 Almond Joy and 13 Twizzlers.(a) You pick a candy at random. What is the probability that you get an Almond Joy?(b) You like neither Twizzlers nor Almond Joy, so you return
You have a box of 40 fuses that look exactly alike except for the colour.There are 4 violet (V), 22 blue (B), 7 red (R), 5 orange (O), and 2 yellow (Y)fuses. Find the probabilities of the following samples.(a) YBBROBROV (sampling with replacement).(b) YBBROBROV (sampling without replacement).(c)
A bag of Christmas candies contains 20 candies in red foil, 12 in green foil, and 8 in blue foil. If you eat 10 candies at random, what is the probability that you eat 5 red, 3 green, and 2 blue?
A bag of 50 identical looking chocolate balls contains 30 chocolates filled with liqueur (L), and 20 filled with marzipan (M). If you eat 10 random chocolates, what is the probability that you eat them in the following sequence: LMLMLMLMLM?
You have flipped a coin 50 times. The probabilities of heads and tails are equal: 1/2.(a) Find P(HTHTTTHHHTHHTTTTTHTTHHH HTTTHHHHTTTTHHHTTTTHHHTHTTT).(b) If we rearrange the Hs and the Ts into a different sequence of 27 tails in 50 attempts, what is the probability of this new sequence?(c) Howmany
You’ve been flipping coins again. This time only 8 times. The probabilities of heads and tails are equal: 1/2.(a) What is the probability of TTHTTTHT?(b) If we rearrange the Hs and the Ts into a different sequence of 5 tails in 8 attempts, what is the probability of this new sequence?(c) How many
You’ve been flipping a coin again, 8 times. But this time the coin is biased, with a probability of heads of 1/3.(a) What is the probability of TTHTTTHT?(b) If we rearrange the Hs and the Ts into a different sequence of 6 tails in 8 attempts, what is the probability of this new sequence?(c) Is
Gold Digger Airlines run a daily shuttle between San Remo and Dry Creek. They have two aircraft: a two-engine DC-3 that used to belong to a Columbian drug cartel, and an old four-engine DC-6B bought from the US Army. All the six engines, that is, all the engines on both planes, have the same
Birthdays: we are investigating to figure out how many persons may be in the same room, before the probability that at least 2 of them have the same birthday exceeds 1/2. Are you able to make a good guess at the answer in advance?(a) Musa and Ibrahim wonder what the probability that the two are not
For each of the official hands in regular five-card hand poker, calculate the number of ways to realise such a hand. (Warning: calculations of the simplest hands are somewhat demanding!
(a) Why was EcoCab not found guilty in Example 6.6.1, even though the eye witness stated that the offending taxi was green (EcoCab’s color)?(b) Is it possible to use Bayes’ theorem when sampling without replacement?(c) As a follow-up to the previous question: “Yes, but…?” – yes, but
Your best friend is holding a black and a white marble, and asks you to open your hands and close your eyes.(a) He deposits one marble in each of your hands, and asks you to close your hands before you open your eyes.What is the probability that the marble in your right hand is black?(b) He asks
Do all of the exercises in a Bayes’ theorem table. Do in addition try out at least one by using the formula directly, to see for yourself that the two methods give the same result. i. Calculate P(B). ii. Calculate all posterior probabilities P(A|B). (a) Prior: P(A) = P(A2)= Likelihood: P(BIA) =
You have 3 urns. In the first urn, there are 91 red and 34 blue balls. The second contains 14 red and 25 blue balls, and the third has 40 red and 25 blue. You pick an urn that random.(a) What are A1, A2, and A3?(b) What are the prior probabilities P(A1), P(A2), and P(A3)?(c) What are the
Text exercises For each exercise:(a) identify the alternatives Ak;(b) identify the pivotal event B;(c) calculate the prior probabilities P(Ak) and the likelihoods P(B|Ak);(d) calculate the posterior probabilities P(Ak|B).i. You have 6 dice, D1, D2, D3, D4, D5, and D6. A friend of yours picks one of
A certain Mr. Claus has mixed up his Scandinavian gift bags this Christmas, and he does not know which one goes to Denmark, which one goes to Sweden, and which one goes to Norway. The Danish bag has 70% soft gifts, the Norwegian one has 40%, whereas the Swedish bag has 20%.Mr. Claus has now put one
Sinterklaas Inc. make two kinds of advent calendar. Type A contains 8 pieces of marzipan and 16 pieces of chocolate. Type G contains 16 pieces of marzipan and 8 pieces of chocolate. They make 3 times as many A calendars as G calendars. You have bought one of their advent calendars, but do not know
It is rumoured that, every year, the Alaskan Easter Bunny brings all children in his state an easter egg filled with 30 chocolates. The eggs are usually filled with 22 milk chocolates and 8 white chocolates, but this year the Mr. Easter has hired some new bunny ladies in the chocolate kitchen, and
A confidential poll by the polling company Giddyup showed that in a certain area 1 in 4 lawyers belong to a secret society; 2 out of 3 lawyers who are members of secret societies had lied to protect a client, whereas: half of all lawyers in general had lied to protect a client. Which proportion of
Assorted candies II: You have 4 bags of mixed candies on the table. They are leftovers from a party, so naturally there are only Almond Joys (A) and Twizzlers (T) left. The contents are as follows:a. A bit tired from tidying up, you zone out and pick a bag at random.You start picking out candies,
Calculate the posterior probabilities for the dice in Examples 6.4.3 and 6.4.2 when you collect all the observations into one big observation for your one update, i.e. WRRR. Compare your one-step updated probability – let’s call it P′1 – to P2 from Example 6.4.3.What do you see?Examples
You have two D6 dice. Die 1 is all white, whereas die 2 is all black. Games master picks one of them. Let Ak be that he picked die k.(a) What are the prior probabilities P0(A1) and P0(A2)?(b) What is the probability that his first toss lands white, p = P(white)?(c) What is the probability that his
Details for Example 6.7.1. You got RRW.(a) Calculate the posterior probabilities P1 for each of the dice.(b) Find p = P1(R), the probability that the next observation is R.(c) If you have sampling with replacement, where the alternatives are red (R) and white (W), and p = P1(red) from the above,
You are selling a car brand that has 2 factories.The factories’ output is equal. Lately, 2.5% of the cars from Factory A have had problems with their brakes, whereas only 0.5% of the cars from Factory B have had such problems. The manufacturer never reveals which factory has produced which cars,
(a) What are the smallest and largest values possible for a discrete distribution function f (x)?(b) What are the smallest and largest values possible for a continuous distribution function f (x)?(c) What are the smallest and largest values possible for a cumulative distribution function F(x)?(d)
In the exercises below, you are given concrete stochastic variables X.(a) X is the outcome of flipping a fair coin with “0” on one side and “1”on the other.(b) X is the sum of the outcomes of two fair coins with “0” on one side and “1” on the other.(c) X is the sum of the outcomes
For the following sets M and functions f , do the following. ~f and Make a table and diagram for f. Determine iff is a discrete probability distribution. If it is, let X ~ do the following. i. Find x. ii. Find Var(X). iii. Find ox. iv. Find TX- v. Find P(X = M). 10 otherwise. if x = 1, 2, 3, 4 if
We create a family of distributions, determined by a parameter r. For each r, let k if n = 1, 2,...,r h,(n) = (n-1)! 0 otherwise
Which of the following functions are continuous probability distributions? 1/40 x [-10,0] f(x)=3/40 x = [0, 10] 0 x [-10, 10] 1/2 x [2,6] f(x)=-1 x = [6,7] 0 x [4,7] f(x)= 0 -sin(x) x [x, 2x] x [, 2] 1/6 x = 4 f(x) = 1/3 x = 5 1/2 x = 6 0 x {5, 6, 7}.
For the following sets M and functions f , do the following in (a)–(d). Graph f, and mark [a, b] along the horizontal axis. Determine whether f(x) is a continuous probability distribution. If it is, let X ~f and do the following. i. Find x. ii. Find Var(X). iii. Find ox iv. Find Tx. v. Find P(X
Find the charge for Sandra’s policy, should the payment be capped at 5000000.
Soren K. has studied previous exams in German philosophy, and has discovered that the lecturer seems to love the two obscure philosophers Max Stirner and Karl Werder. Soren sets up a table he believes expresses the joint probabilities for the respective number of questions on Stirner (X)and
You have a bag of one each of the dice D4,D6,D8,D10,D12,D20. You sample a die, and toss it. Let X be the number of faces on the die, and let Y be the value of the toss. Are X and Y independent?
X follows a discrete uniform distribution over the \(n\) numbers \(\{a, a+1\), \(\ldots, b-1, b\}\) if \(P(X=c)=1 / n\) whenever \(c\) is in the list, and 0 otherwise. Use the rules from chap. 7 to answer the problems below.(a) Find \(\mu_{X}\).(b) Find \(\sigma_{X}^{2}\).(c) Sketch the probability
The stochastic variable \(X\) follows a uniform probability distribution over the integers \(\{0, \ldots, 99\}\). What, then, is \(P(X \in\{2,3,5,7,11,13,17,19\})\) ?
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