Description
Part I: Calculations and derivations by hand
1. (10 points) Bob and Carl have just learned the Law of Large Numbers. It turns out that they
have a different understanding of what the law says.
Bob: The Law of Large Numbers says that in the long run, a fair coin will land heads as often
as it lands tails.
Carl: I don’t think that’s what it says. The Law of Large Numbers says that the fraction of
heads will get closer and closer to 1/2, which is the expected value of each toss.
Bob: Isn’t that the same as what I said?
Carl: No, you said, “The coin will land heads as often as it lands tails,” which implies that the
difference between the number of heads and the number of tails will get smaller as we toss the
coin more and more. I don’t think the number of heads will be close to the number of tails.
Bob: If the fraction of heads is close to 1/2, then the number of heads must be close to the
number of tails. How could it be otherwise?
Who is right: Bob or Carl?
(1) Calculate the variance of
number of heads in 𝑛 tosses − number of tails in 𝑛 tosses
as a function of 𝑛. (5 points)
(2) Considering this calculation, do you agree with Bob that the difference between the
number of heads and the number of tails approaches 0 as the number of tosses increases?
(5 points)
2. (15 points) Let 𝑋!, 𝑋”, … , 𝑋# be a sequence of independent and identically distributed random
variables. Define 𝑌# = (𝑋! + 𝑋” + ⋯ + 𝑋#)/𝑛. Show that the sequence 𝑌!, … , 𝑌# converges
in probability to some limit, and identify the limit, for each of the following cases:
(1) 𝑋$ follow the Poisson distribution with parameter 3. (5 points)
(2) 𝑋$ are uniformly distributed over [−1,3]. (5 points)
(3) 𝑋$ follow the exponential distribution with parameter 5. (5 points)
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3. (15 points) Continued with Example 4.5 (3), prove that if 𝑈!~U[0,1] and 𝑈”~U[0,1] are
independent, and let
G
𝑍! = I−2 ln(𝑈!) cos(2𝜋𝑈”)
𝑍” = I−2 ln(𝑈!) sin(2𝜋𝑈”)
,
then 𝑍! and 𝑍” are a pair of independent standard normal random variables.
(Hint: Show that𝑃(𝑍! ≤ 𝑎, 𝑍” ≤ 𝑏) = Φ(𝑎)Φ(𝑏) for all 𝑎 and 𝑏, which requires variable
substitution in a double integral.)
4. (10 points) Explain how to generate numbers from the following distributions based on a
uniform distribution random number generator:
(1) Geometric(p); (5 points)
(2) Standard Cauchy distribution, of which the PDF is 𝑓(𝑥) = 1/[𝜋(1 + 𝑥”)]，𝑥 ∈ (−∞, ∞).
(5 points)
Part II: Implementations by Python
Note: Please provide both the results and the code, and present them in one PDF file.
5. (15 points) Continued with Problem 4, generate 10000 numbers from the following
distributions based on a uniform distribution random number generator in Python, plot the
histogram of the generated numbers, and compare it with the theoretical PMF/PDF of the
following distributions:
(1) Geometric(0.5); (5 points)
(2) Standard Cauchy distribution. (10 points)
6. (15 points) Apply the rejection sampling technique to sample from
𝑓∗(𝑥) = 0.6 exp{−(𝑥 + 5)”/2} + 0.4 exp{−(𝑥 − 1)”/0.5}, 𝑥 ∈ (−∞, ∞).
(1) Choose an appropriate proposal distribution and plot it to show that it covers the target
distribution. (5 points)
(2) Apply the rejection sampling method to generate 500,000 samples, plot the histogram of
the generated samples, and compare it with the theoretical PDF 𝑓(𝑥) (normalized 𝑓∗(𝑥)).
(5 points)
(3) What is the acceptance proportion? (5 points)
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7. (20 points) A forest consists of 1,000 trees forming a perfect 20 × 50 rectangle as in the
figure below. The northwestern (top-left) corner tree catches fire. Wind blows from the
northwest, therefore trees can only catch fire from its buring left and above neighbors (⽕只
会向东或向南蔓延). Assume the probability that any tree catches fire from its burning left
neighbor is 0.8, and the probabilities to catch fire from trees immediately to the above is 0.3.
(1) Conduct a Monte Carlo study to estimate the probability that more than 30% of the forest
will eventually be burning. With probability 0.95, your answer should differ from the true
value by no more than 0.005. (10 points)
(You should first determine the value of 𝑛 that satisfies the requirement, and then
conduct a Monte Carlo study to estimate the probability.)
(2) Based on the same study, predict the total number of affected trees 𝑋. (5 points)
(3) What is the corresponding standard deviation of 𝑋. (5 points)
