概率代考-Prob 140代写-Probability考试代写
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概率代考-Prob 140代写-Probability考试代写

Prob 140 Fall 2019

MIDTERM EXAM

概率代考 Please leave all finite sums unsimplified, whether they are numerical or algebraic, umnless the question requires a simplification.

NAME (FIRST LAST):————————–  SID:——————

BUILDING (circle one): VLSB/Dwinelle    SEAT NUMBER:—————

TIME AND CONDITIONS: 75 minutes; closed book/ notes/internet; no calculator/computer

QUESTIONS AND ANSWERS

●There are 6 questions.

●Give brief explanations or show calculations in each part of each question unless the question says this is not required. You may use, without proof, any result proved or 1used in lecture, the textbook, homework, and, labs, unless the question asks for a proof.

●You may answer any, part of any question, If the answer to one part depends on another that you couldn’t do, you can still provide an answer such as“The answer to part (a), divided by 2.”

●Please simplify all infinite sums. Please leave all finite sums unsimplified, whether they are numerical or algebraic, umnless the question requires a simplification.

GRADING  概率代考

●The exam is worth 50 points.’ Each of the first 5 questions is worth 8 points. Question 6 is worth 10 points. Points for parts are indicated in the questions.

●We will give partial credit, but only for substantial progress towards a correct answer. We get to decide what“substantial progress”means.

●Please cornmit yourself to a single answer for each part of each question. If you give multiple answers (such as both True and False), please don’t expect credit, even if the right answer is among those that you gave.

1. There are 100 particeipants in a randomized controlled experiment. Participants are assigned to the treatment and control groups in a way that ensures that there are at least 20 partcipants in each group.The randomization process is 88 follows.

●Stage 1; A fair coin is tossed for each participant. If the coin lands heads the participant is assigned to the treatment group, and if it lands tails the participant is assigned to control. If after all 100 tosses there are at least 20 participants in each group, the randomization is complete. If not, the process continues to Stage 2.  概率代考

●Stage 2 (if needed): If the randomization is not complete at Stage 1, the process. begins afresh.Participants are assigned to treatment and control based on pew fair coin tosses as described above. If after all 100 tosses there are at least 20 participants in each group, the randomization is complete. If not, the process continues to Stage 3.

●Subsequent Stages (if needed): The process continues in this way, one stage at a time, until the randomization is completed.

a) [5 points] What is the chance that the randomization is completed in Stage 1?

b) [3 points] Let R be the number of stages required for the fandomization to be completed2 Find E(R).

2. Let Xo, X1, X2… be a Markov Chain with time-homogeneous one-step transition probabilities displayed the diagram below. Suppose P(Xo= 1)= 1.

a) [2 points] Is there a probability distribution π that satisfies the detailed balance equations for this chain? Why or why not?

b) [2 points] Fill in the blank with a umber and show your calculations:

In the long run, the expected proportion of time the chain spends in State2 is——times the expected proportion of time the chain spends in State 1.  概率代考

c) [4 points] In which of the four states do you think X500 jis most likely to be? Justify your answer with calculations.

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3. At a grocery store, the number of people in line at the checkout counter at 10 a.m, depends on the day of the week.

●On each weekday (Monday, Tuesday, Wednesday, Thursday, and Friday) the number of people in line at 10 a.m. has the Poisson (3) distribution, independently of all other days.

●On each weekend day (Saturday, Sunday) the number of people in line at 10 a.m.? has the Poisson (6)distribution, independently of all other days.

Each day, each person in line brings their own grocery bag with chance 0.9, independently of all other people and days.  概率代考

Suppose you pick one of the seven days next week at random, independently of the people in the store. Let G be the number of people who are in line at 10 a.m, that day and bring their own grocery bag.

a) [4 points] Find E(G)..

b) [4 points] Find P(you pick ThursdayIG = 5).

4. A bowl of Halloween candy contains 40 Fun Size Snickers bars. There are 10 bars of each of four different kinds of Snickers: Original, Peanuit Butter, Almond, and Crisper. Bars are drawn out one at a time at random without replacement. Let D be the number of draws till all four kinds have appeared.  概率代考

a) [4 points} For eachnf 0, 1,2…. find-P(D> n)

b) [2 points] Find E(D)

c) [2 points] Find the chance that the kind of bar that appears on Draw D is Original.

5. Let X have the Poisson (μ) distribution.

a) [3 points] Find

b) (5 points] Find

[Hint:) First simplify the diference for a non- negative integer n.]

6. Let X1, X…..Xn be a random permutation of 1.2.3….n. That means the sequence X1.X….n is equally likely to be any ofthe n! permutations of 1,2.3…,n. Here are two features of random permutations. You encountered the first one in your code-breaking lab.

●Transpositions: There is a transpositionat a pair of indices i≠j if Xi=j and Xj=i.

●Records:r There isa record at an indexi > 1 ifX;> Xk for allk= 1,2….i- 1. To take care of the edge case, the definition also says that there is always a record at index i = 1. For example, in the permutation 253461978 of the integers 1 through 9, the records are at indices 1, 2, 5, and 7.  概率代考

a) [4 points] Find the expected number of transpositinis in the random permutation X1,X2,……,Xn.

b) [4 points] Fix an integer i > 1. Find the chance that the random permutation X1,X2,….,Xn has a record at index i.

c) [2 points] Find the expected mumber of records in the random permutation Xi, X2,…..,Xn.

 

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