answer the following

Consider the symmetric simple random walk at two timepoints: n and n + m. Find

?(Sn, Sn+m). What happens as m ? ? for fixed n and as n ? ? for fixed m?

Explain intuitively.

23 Consider the simple random walk with p 6=

1

2

. Use the law of large numbers to argue

that Sn goes to ?? if p

2

and to ? if p > 1

2

.

24 Consider a symmetric simple random walk with reflecting barriers 0 and a, in the sense

that p0,0 = p0,1 =

1

2

and pa,a?1 = pa,a =

1

2

. (a) Describe this as a Markov chain

and find its stationary distribution. Is it the limit distribution? (b) If the walk starts

in 0, what is the expected number of steps until it is back? (c) Suppose instead that

reflection is immediate, so that p0,1 = 1 and pa,a?1 = 1, everything else being the

same. Describe the Markov chain, find its stationary distribution ?, and compare with

(a). Explain the difference. Is ? the limit distribution?

25 Consider a variant of the simple random walk where the walk takes a step up with

probability p, down with probability q, or stays where it is with probability r, where

p + q + r = 1. Let the walk start in 0, and let ?1 be the time of the first visit to 1. Find

P0(?1

26 Consider the simple random walk starting in 0 and let ?r be the time of the first visit to

state r, where r ? 1. Find the expected value of ?r if p > 1

2

.

27 Consider the simple random walk with p 6=

1

2

, starting in 0 and let

?0 = min{n ? 1 : Sn = 0}

the time of the first return to 0. Use Corollary 8.3.2 to show that P0(?0

2 min(p, 1 ? p).

28 Consider the simple random walk with p > 1

2

started in state 1. By Corollary 8.3.2

"reversed," the probability that the walk ever visits 0 is (1 ? p)/p. Now let the initial

state S0 be random, chosen according to a distribution on {0, 1, ...} that has pgf G. (a)

Show that the probability that 0 is ever visited (which could occur in step 0 if S0 = 0)

is G((1 ? p)/p). (b) Now instead consider the probability that 0 is ever visited at

step 1 or later. Show that this equals G((1 ? p)/p) ? 2p + 1. (c) Let p =

2

3

and

S0 ? Poi(1). Compute the probabilities in (a) and (b) and also compare with the

corresponding probability if S0 ? 1.

29 Consider a three-dimensional random walk Sn where in each step, one of the six neighbors along the axes is chosen with probability 1

6

each. Let the walk start in the origin

and show that

P(S2n = (0, 0, 0)) =

1

6

2n X

i+j+k=n

(2n)!

(i!j!k!)2

and use Stirling's formula to conclude that the walk is transient.

30 Consider a branching process with mean number of offspring , letting Yn be the total

number of individuals up to and including the nth generation and letting Y be the total

number of individuals ever born. (a) For what values of is Y finite? (b) Express Yn

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. p ( x ) = / voor g (x) = Need Help? Which It Tam to a Tudor 4. -1 points SCakET8 5 3.008 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. Need Help? S. . =1 paint SCocCT8 5.3011. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. F(x) = Need Help? Ford Watch Tied to a Tutor 4 +1 points ScottTIS3013 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.Using the following income data, calculate a ttest statistic to determine if the difference between the twogroup means is statistically signicant Hint: Unequal Population 1'afariance Assumed. mm w\"m Assumptions: State your Research and Null Hypothesis: Selecting the Sampling distribution and Specify the Test Statistic: Compute the Test Statistic: Show your work: Make a Decision and Interpret the Results: instructure.com/courses/12354/assignments/259572 Does it take less time for seeds to germinate if they are near rock music that is continuously playing compared to being near clausical mulch The 49 weeds that were exposed to rock music took an average of 25 days to germinate. The standard deviation waa 15 days, The 52 wench that were exposed to clayical music look he average of 20 days to germinate. The standard deviation for these torch was 3 days. What can be concluded at the a = 0.01 level of significance? a. For the study, we should use Select on myem b. The null and alternative hypotheses would be: Fo: Select an andwer Select an ameer B) Select an meme B (lette enter a decimall Srest in sower B (Please enter a decimal) C. The test statistic +8- (pleme show your answer to 3 decimal places. ] d. The prvalue = (Pleme show your answer to 4 decimal places. ) e. The p-value is Ba I, Breed on this, we should Salici an answer ] the null hypothesis, . Thus, the final conclusion is that .. The results are stathically significant at cr = 0.01, so there is sufficient evidence to conclude that the mean germination time for the 49 seech exposed to rock music that were observed is less than the mean germination time for the 52 weech that were exposed to classical music that were observed. The rewelts are Mathitically significant at a = 0.01, so there is willcient evidence to conclude that the population mean time for weeds exposed to rock music to germinate is less than the population mean time for preds co posed to classical musk to germinate. The results are statistically insignificant at o = 0.01, so there is statistically significant evidence to conclude that the population mean time for seeds exposed to rock music to germinate is equal to the population mean time for seeds exposed to classical music to germinate. The rewits are pathetically insignificant at a = 0.01, so there is Imufficiant evidence to conclude that the population mean time for weech exposed to rock muise to germinate is Less than the population mean time for seeds exported to classical music to germinate. h. Interpret the p-value in the context of the Mucly. There is a 4.05k chance of a Type I error. There Is a 4.03% chance that the mean germination time for the 49 seeds exposed to rock music is at least 4 days Wis than the mean germination time for the 51 weech exposed to classical music. the population mean time for seeds exposed torock music to germinate is the same as the MacBook Air 3 W E R T Y U O P D F G H K X C V N MUse part I of the Fundamental Theorem of Calodus to find the derivative of Given #- 1 d't At what value of r does the local max of f(x) occur