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Homework 1 hints

I've just answered some questions about the homework, and in the interest of fairness, here are the answers for everyone.

  • In question 2.3 onwards (the questions about the coin) the discussion is asking about probability densities rather than probability distributions. For the purposes of Bayes' rule you can treat them the same way. The only distinction is that when normalising a probability density you integrate rather than summing. Similar to probability distributions, probability densities must integrate to 1.
  • In question 2.9 you should note that the probabilities are all combined using multiplication. This means that everything is either independent or conditionally independent.
    • If x is normally distributed, then the probability that x is between a and b is P(a < x < b) = \int_{x = a}^b \mathcal{N}(\mu, \sigma^2, x) dx.
    • If two probability distributions are independent then this integral decomposes nicely. If x and y are conditionally independent, P(x,y|z)=P(x|z)P(y|z), then the probability x is between a and b and y is between c and d decomposes: \int_{x = a}^b \int_{y = c}^d P(x,y|z) dy dx = \int_{x = a}^b P(x|z) dx \int_{y = c}^d P(y|z) dy.
    • P( A or B ) = P(A) + P(B) - P(A intersects B). When A and B are disjoint (they can't both be true at the same time) then P(A intersects B) is 0 and so P( A or B ) = P(A) + P(B).
    • Also, I've come across a small bug in the question (an 'off by 1' error) which means that the probabilities would be rather small if you worked them out. I'm going to leave the question as it is however - the mathematics is the same, even if we're now talking about somewhat improbable events.