Part 1: Consider a "random rotation" of a fixed point on a unit sphere in p-dimensions, where p is at least 3. (For example, you're standing near the Washington Monument, and the next thing you know you find yourself in Paris.) A move to Paris from DC may be viewed as an "orthogonal transformation". Such a transformation is represented by a 3 x 3 orthogonal matrix, and may be regarded as a "rotation" from one place to another. If the initial rotation from DC is random and uniformly distributed (I'll explain what I mean by this), you could have ended up anywhere. "in the sense that your location is uniformly distributed on the surface of the unit sphere. (That you happened to end up in Paris is pure luck!) Now, suppose that the same transformation is used to rotate you again to a new place on the unit sphere. What can you say about where you are now? (Have you moved back closer to DC, for example, or could you still be anywhere?) And what if this keeps happening to you, again and again?
Part 2: Here's a simply worded statistical problem, but one for which a really satisfactory solution seems elusive. Given a random sample from a k-variate normal distribution with unknown mean vector and unknown covariance matrix, produce a confidence interval for the maximum component of the mean vector. I'll motivate this problem by describing what a regulatory guidance says should be done when analyzing data collected in a "thorough QT study", and describe an approach which works well in some circumstances. (Q and T are two points on an electrocardiogram (ECG), and the duration of the interval between them, the 'QT interval' represents the working phase of the heart -- when the heart is contracting. In the pharmaceutical industry, studies that indicate significant QT prolongation are often sufficient for a company to discontinue development of a compound.)