What's the point of transforming variables in engineering?

E.g. in video, (R,G,B) might be transformed to (Y,I,Q) with a 3x3 matrix multiply. Y is brightness, I,Q are color. Since we see brightness more accurately than color, we want to transmit Y with greater precision. So, we want to do probabilities on all this.

Exercise 5.25 p291

Exercise 5.37

The number of full pairs is the floor of half the number, i.e., (1,2,3,4,5,6)->(0,1,1,2,2,3).

Exercise 5.56

Max and min of 2 uniform random variables

1. W=min(X,Y). Z=max(X,Y).
2. cdf(W), cdf(Z), pdf, mean
3. Probability that a 3rd r.v. will be bigger than the first 2.
4. Quantiles of of n r.v.
5. What is the probability that the max of 3 uniform (0,1) random variables is greater than .5?
   • A: 1/8
   • B: 1/3
   • C: 1/2
   • D: 2/3
   • E: 7/8

5.8.3 Transformation of 2 r.v.

1. X, Y are exponential with {$\lambda=1$}.
2. What is joint pdf of (X+Y, X-Y)?