Questions of Essence

A collection of questions which make good exercises.

• Let \(X\) be a random variable with probability density \(P(X)\), \(G\) a measurable function. It is easy to see that \(G(X)\) is a random variable as well, and suppose it has density \(P(G(X))\). Now suppose \(P(X,G(X))\) is the joint distribution over these 2 random variables. Justify whether or not \(\int P(X,G(X)) \: dX = P(G(X))\).
• Given an urn with \(n\) green and red balls, where green ball has a diameter uniformly chosen between \([g_1,g_2]\) and each red ball has a diameter uniformly chosen between \([r_1,r_2]\). Consider the case \([g_1,g_2]=[1,2], [r_1,r_2]=[1,3]\), what the probability that the colour is green given that one chooses a ball with a diameter of 1.5?