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?

Thoughts

Created: 2022-03-13 Sun 21:45

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