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Table Of Contents
Mathematics of Machine Learning
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Problem 1. Let X and Y be two independent random variables, and let a,b ∈ℝ be two arbitrary constants. Show that X −a and Y −b are also independent from each other.
Problem 2. Let X be a continuous random variable. Show that P(X = x) = 0 for any x ∈ℝ.
Problem 3. Let X ∼ Bernoulli(p) and Y ∼ Binomial(n,p). Calculate the probability distribution of X + Y .
Problem 4. Let X ∼ Bernoulli(p) be the result of a coin toss. We select a random number Y from [0,2] based on the result of the toss: if X = 0, we pick a number from [0,1] using the uniform distribution, but if X = 1, we pick a number from [1,2], once more using the uniform distribution. Find the cumulative distribution function of Y . Does Y have a density function? If yes, find it.
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