CSCI 2244 Randomness and Computation - PS4: Conditional Probability, Poisson Distribution and Benford’s Law
CSCI 2244.02 Fall 2023 PS4 Instructor: Shang-En Huang
Problem Set 4
Release Date: September 20, 2023. Due at: 10pm EDT, September 27, 2023.
General Instructions. The purpose for having these problem sets is to enhance your mathematics and programming skills when it comes to the probability theory. These problems may be quite challenging, so discussions with classmates are encouraged. However, I strongly suggest you spend at least half an hour thinking on the problems individually before discussing. Although discussions (and consulting to ChatGPT and WolframAlpha) are allowed, you need to write the solution using your own words by yourself. Please acknowledge any person or references that you discuss with or consult from.
1 Conditional Probability (4 points)
Consider the random experiment that keep tossing a coin forever. For any integer k, l ∈ N. Let Xk be the total number of coins tossed whenever the k-th HEAD shows up. Let Sl be the total number of HEADs during the first l coin tosses.
(a) (b)
(c) (d)
(1 point) For any two integers s and t such that 1 ≤ s < t, find (and explain) the conditional probabilityP(X1 =s|X2 =t).
(1 point) For any two integers s and t such that 1 ≤ s < t, find (and explain) the conditional probability P (X100 = t | X99 = s).
(1 point) Find (and explain) P (S100 = 5 | X5 = 100). (1 point) Find (and explain) P (X5 = 100 | S100 = 5).
2 Poisson Distribution (4 points)
Let X be a Poisson random variable with parameter λ.
(a) (3 points) Let t ≥ 0 be an integer. What value of λ maximizes the probability P (X = t)? Find and explain your answer.
(b) (1 point) Show that P(X is even) = 1(1 + e−2λ). (Hint: How did we prove P n = 2n−1?) 2 k≥0 2k
3 Other Distributions (4 points)
(2 point) (Similar to Banach’s matchbox problem) You have two empty urns, and each urn has a capacity of exactly N balls. Each day you throw a ball into one of the urn, uniformly at random. On the day where exactly one of the urn becomes full, what is the distribution on the number of balls in the other urn?
(2 points) Using the definitions of Xk and Sl from the first problem, argue that for any pair of integers k and l:
P(Xk >l)=P(Sl <k).
Remark: Xk is actually a negative binomial distribution and Sl is actually a binomial distribution.
Weekly Quiz 4 (4 points)
Please complete weekly quiz 4 on Canvas.
Programming Assignment 4: Benford’s Law (4 points)
Please complete PA4.ipynb and submit it to Canvas.
