• No extension will be provided, unless for serious documented reasons.
• Start early!
• Study the material taught in class, and feel free to do so in small groups, but the solutions should be a product of your own work.
• This is not a multiple choice homework; reasoning, and mathematical proofs are required before giving your final answer.
• The code necessary for problem 2 should be written in the Jupyter notebook handed out to you.
1 Probability [25 points]
Solve the following problems:
a. (5pts) Let X,Y be independent random variables with common density function f. Prove that the density function of Z = max(X,Y ) is given by fZ(x) = 2f(x)P(X ≤ x).
b. (10pts) If U is a uniform random variable in [0,1], what is the distribution of ⌊100U⌋+1?
c. (10pts) If U is a uniform random variable in [0,1] and 0 < q < 1, prove that X = has a geometric distribution. What is the parameter of the geometric
distribution?
2 Bayes rule [20 points]
Let N be a discrete random variable that takes values from the set {1,n} with equal probability, i.e., Pr . Consider the following process.
• First we draw a value for N.
• Then, we draw N iid uniform RV {Xi}i=1,...,N in [0,1].
Someone tells you the value Z = mini=1,...,N Xi = 0.05, namely the smallest value among the N uniform RVs drawn. However you do not know the value of N.
3 Needles and Probability [55 Points]
The Jupyter notebook in on Git.
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