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Extra info for A tutorial on Principal Component Analysis

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Suppose that we want to distribute ﬁve numbered balls over three boxes I, II and III. Each ball is put in a random box, independently of the other balls. Describe an appropriate sample space and probability measure for this 32 Chapter 1. Experiments experiment. Compute the probability that (a) box I remains empty; (b) at most one box remains empty; (c) box I and II remain empty. 26. An urn contains 10 white, 5 yellow and 10 black balls. We pick a random ball. What is the probability that the ball is yellow, given that it is not black?

For instance, when the random variable X satisﬁes E(X) = ∞, and Y = −X, then E(X) + E(Y ) is not deﬁned, but E(X + Y ) = 0. 2. 9 should only be true when X and Y are independent. However, the preceding calculation shows that the result has nothing to do with independence. 10. 9 to more than two random variables. 3. 11. For any random variable for which E(X) exists and for any a and b, it is the case that E(aX + b) = aE(X) + b. 12. Prove this proposition. Instead of sums, we also need to consider products of random variables.

We will now explain why method (1) is wrong. 2). This seems obvious, but is, in fact, not correct. The fact that the ﬁrst person to be checked has the particular DNA proﬁle, says something about the total number of individuals with this proﬁle. 6. In that example, even when we know that a family has at least one boy, when we then actually see a boy opening the door, this new information does change the conditional probability that the family has two boys. The bare fact that a boy opened the door, makes it more likely that there are two boys.