Statistics - Stochastic Assignment Help

Statistics  Stochastic Assignment Help

Assignment Task:

This is a stochastic assignment! Before beginning, please have at your disposal (a) a fair coin with identifiable "heads" and "tails", (b) a fair six-sided die with faces numbered one through six, (c) a pen of your choosing, and (d) a printed copy of this assignment. 

Okay... are you reading this in print and ready with your coin, die, and pen? Please proceed according to the following rules. 

•  Whenever you see 4, flip your coin and, in the space provided, write down 1 if you  throw "heads" and 0 if you throw "tails”. 
• Whenever you see ?, roll your die and, in the space provided, write down your rolled number from the set {1,2,3,4,5,6}. 
• After filling in all of the A and ? spaces, carry out any remaining arithmetic operations. 

You should now have a realization of this stochastic assignment. Now, please attempt your questions, and don't forget to attach your stochastic assignment sheet when you land it in!

1. Let (N4, t > 0) be a Poisson counting process with rate 1 = __; and define 

t1 = 2 * = —, t2 = t? +(1+2) ==—, n1 = =+==~, and n2 = nitoto 

(a) Determine P(Nt1 = N1, Nt, = n2). 

 (b) Determine P(N(1, t?) = n1, N(ta - 1, 42] = na). 

 (c) Determine P(N(1, ti] = ni/N(t? – 1, ta] = n2). 

 (d) Determine EN(1, t?]. 

 (e) Determine E (N(1, t?]|N(t? – 1, ta] = n2

(f) Determine E (N(t? – 1, t2] [N(1, t?] = n?). 

 (g) Show that, for 0 < s <t and n>0, (N,|N= n) ~ Bin(n, s/t). [1] (h) Denote by Tk the time of the k-th event. Show that, given Ni = 1, Ti is uniformty distributed on 0,1]. (i) Determine the pdf of Ti given Ni = 2. (j) * Calculate E[Ti N1 = n) for n=0,1,2,.... (k) # Calculate E[T2N1 = n) for n = 0,1,2,.... (1) * Suppose now that (N+, t > 0) is a non-homogeneous Poisson counting process with rate function /(t) = 2+_+(-1) + x sin (_Xaxt+_x1) =  _, for t > 0. Repeat (a)-(f). XT 

(m) # In the context of (1), determine the pdf of T? given N1 = 2. (n) # In the context of (1), determine the distribution of (N. Nt = n) for 0 <s and n = 0,1,2, .... 

2. The Poisson café is well-known for its apple pie and blueberry muffins and trades  between 10 am and 4 pm. When the café is open for custom, patrons arrive according to a Poisson process with constant rate 1 = 10 x = _ . per hour.  Suppose that each arriving patron's food order is: an apple pie with probability P1 = ___/24 = ___, or a blueberry muffin with probability P2 = _ /24 = _ ,  or an apple pie and a blueberry muffin with time-dependent probability pz(t) =  o * exp(10 – (1+_) * t)/24 =  -, for t E (10,16] (i.e., hours), or neither an apple pie nor a blueberry muffin with probability p4(t) = 1- P1 – P2 – P3(t) =

(a) On a given trading day, what is the probability that there are (2+_  __ food orders for only an apple pie between 10 + __ = _  )x= = am and  1+2 x __= __ pm?  _ 

(b) Repeat (a) but now for any food order with an apple pie. 

 (c) On a given trading day, what is the probability that there are 3 x — =.  food orders for only a blueberry muffin between 10 + --- = _ am and 1+3 x — = — pm, 2 x — = — food orders for both an apple pie and a blueberry muffin between 10+ __ = _am and 1+_ = ___pm, and ___ = _ patrons who order neither an apple pie nor a blueberry muffin between 10+ __ = am and 1 +2 *_ =_pm? 

(d) * Given that there are 2 x — = — patrons who order neither an apple pie  nor a blueberry muffin on a particular day, what is the expect number of apple pies and blueberry muffins that were ordered on that day? . (e) # Adapting the procedure outlined in class, simulate the food orders of patrons to the Poisson café on a given trading day, making sure to provide a plot that clearly identifies each of the four types. Include you plot and your code. 

 

3. The Sturt Stony Desert is a "gibber” desert partly located in south-western Queens  land (Australia) with an estimated area of 29750 km2. Suppose that you happen to be camping with friends and want to take a picture of the fat-tailed dunnart (Sminthopsis crassicaudata). One of your friends claims that its appearance in this area follows a spatial Poisson process with constant rate 5 x +0.___ = _  per km2. You'd really like to capture a spectacular photo of this unusual desert creature, but before setting out you want to make sure you have a reasonable chance of success if your friend's claim is correct. You and your friend can carry enough  water and food as well as your photographic gear to scour a 0.25 km ? area before you have to return to camp. 

(a) If your friend is correct, what is the probability that you would see at least  one fat-tailed dunnart on a single trip? 

 (b) How many trips (visiting distinct areas) should you expect to take before you  have (2+  = _ sightings of the fat-tailed dunnart? 

 (c) Suppose a recent survey of put the total population of the fat-tailed dunnart in the Sturt Stony Desert at __ 10%. Given this information, what is the conditional probability that you would see at least one fat-tailed dunnart on a single trip? * Suppose now that your friend also claims that if you see a fat-tailed dunnart, you're more likely to see another one nearby. Is this consistent with your friend's earlier claim? Argue why or why no

4. # Let (Nt, t > 0) be a Poisson process with constant rate > 0, X1, X2,...  N(u,02), and St = X1 +...+XNwith So = 0. The stochastic process (St,t> 0) is an example of a compound Poisson process, which is a Poisson process whose unit jumps at event-times are replaced by a jump drawn from some distribution. 

(a) Determine E[St] and Var(St) for all t> 0.

(b) Determine P(St > Ilu tolt) for all

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