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STAT433/833 Assignment 1
发布时间:2023-10-23
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ASSIGNMENT
STAT433/833 Assignment 1:
Due October 24th before class
1 | Textbook Problems [25]
Solve the following exercises from [Res92]:
a) 3.4 [2]
b) 3.7 [4]
c) 3.9 [2]
d) 3.10 [3]
e) 3.19 [3; The process being “Poisson” means that the arrival times are exponentially distributed.]
f) 3.21 [5]
g) 3.46 [3]
h) 3.50 [3]
2 | Size Bias and the Inspection Paradox [25]
Do you ever feel like you wait longer than average for the bus, or that your friends have more friends than you? Well, you’re not crazy. Let’s prove why!
These phenomena are due to something known as size-biasing, or (in the context of renewal theory) the inspection paradox. This phenomena is the reason we needed to use a truncation argument in the proof of [Res92, Theorem 3.3.3]. This occurs whenever our propensity to measure something correlates with the value we measure.
a) Suppose that arrivals times are Yj ∼ Ber(0.5) and r(Y0 = 0) = 1. What is the distribution of YN (0 .5) ? Is it typically at least as large or at least as small as Y1? [2]
b) The transit authority advertises that average time between buses is 5 minutes. Suppose that the time between buses is 0 minutes with probability 80% and 25 minutes with probability 20% – that reflects my experience anyway. Is their claim that the average time between buses is 5 minutes accurate? What is the typical length of the interval between buses for the intervals when I arrive at the stop, assuming I arrive at a random time with some continuous distribution? [1]
c) Suppose that arrivals times (Yj)j≥1 are distributed with CDF F and r(Y0 = 0) = 1. Let [x]+ = max(x, 0).
Show that the CDF of YN(t) is
P(YN(t) ≤ x) = lu∈[0,t][F (x) − F (t − u)]+ U(du) (1)
(Hint set up a renewal equation for Hx(t) = P(YN(t) ≤ x).) [3]
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d) Show that when Yj ∼ Exp(α) then
P(YN(t) ≤ x) = 1 − exp(αx) − α min(x, t) exp(−αx). (2)
How does this compare to P(Y1 ≤ x)? Does this mean YN(t) is typically larger or smaller than Y1? [2]
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e) Show that when Yj ∼ Exp(α) then
E[YN(t)] = α . (3)
How does this compare to EY1? [2]
f) [Maybe harder than the rest!] Show that for general F , P(YN(t) ≤ x) ≤ P(Y1 ≤ x) for all x. This means
that Y1 is stochastically dominated by YN(t) for all t. [3]
g) Show that 101 YN(u ·t)du converges to E[Y12]/E[Y1] with probability 1 as t → ∞. How can we interpret this fact in terms of the length of the renewal interval experienced by an observer arriving at a random time? [4] h) Is lim t→∞ 101 YN(u ·t)du larger or smaller than limn→∞Sn/n? [2]
i) Show that if the arrival distribution F is absolutely continuous, with density f, then
tP(YN(t) ≤ x) = 10x
)ds . (4)
How does this compare to F (x)? [2]
j) Show that if the arrival distribution F is absolutely continuous, with density f, then EYN(t) → E[Y12]/E[Y1]. How does this compare to EY1? [2]
k) Suppose that your social network is represented by a graph 武 = (小 , 含), and that everyone in the network has at least one friend. The number of friends of person i ∈ 小 is deg(i). Recall that Σi∈小 deg(i) = 2 |含 | . i)
Show that the average number of friends a person has is 2 || 小(含)|| . ii) Show that they average number of friends
that a person’s friends has is
Suppose that the friend graph is not d-regular for any d (so that there is some variability in the number of friends each person has). Which is bigger, the average number of friends or the average number of friends of one’s friends? Comment on the similarity between this and the solutions to parts g) and j). [2]
Fixed a typo in Eq. (5)
3 | Grad Reading List
Required of graduate students only!!!
Pick one (1) of the following to read and then write a short (≲ 1 page) summary of, focusing on how it applies or relates to renewal processes.
1. [Res92, Sections 3.7.1 and/or 3.12] and W. L. Smith. “Regenerative stochastic processes” . Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 232.1188 (1955), pp. 6–31 https://www.jstor.org.proxy.lib.uwaterloo.ca/stable/99680
2. [Res92, Sections 3.7.1 and/or 3.12] and S. Stidham. “Regenerative processes in the theory of queues, with applications to the alternating-priority queue”. Advances in Applied Probability 4.3 (1972), pp. 542–577 https://www.jstor.org.proxy.lib.uwaterloo.ca/stable/1425993
3. R. Pyke. “Markov renewal processes: definitions and preliminary properties” . The Annals of Mathemat- ical Statistics 32.4 (1961), pp. 1231–1242. issn: 00034851 http://www.jstor.org.proxy.lib.uwaterloo.ca/ stable/2237923
4. [Res92, Sections 3.5.1] and O. Thorin. “Probabilities of ruin” . Scandinavian Actuarial Journal 1982.2 (1982), pp. 65–103. doi: 10.1080/03461238.1982.10405105. eprint: https://doi.org/10.1080/03461238. 1982.10405105 https://www.tandfonline.com/doi/epdf/10.1080/10920277.1998.10595723?needAccess=true 5. [Res92, Sections 3.5.1] and S. Asmussen and T. Rolski. “Computational methods in risk theory: a matrix- algorithmic approach” . Insurance: Mathematics and Economics 10.4 (1992), pp. 259–274 https://www- sciencedirect-com.proxy.lib.uwaterloo.ca/science/article/pii/016766879290058J