## III. HOME WORK PROBLEMS

### 6. GOAL SOFTENING (2)

Consider the same question as above, except now the population is finite [i = 1, 2, ..., N] with Ji = i. But we must observe Ji through N(0,1) additive noise, i.e.,

Jobserved = Jactual + w
where w ~ N(0,1).

1. What is the probability that if we choose the observed top value out of the population it will turn out to be indeed the true top value?

2. What is the probability that if we choose the 2 observed top values, at least one of them will turn out to be among the actual top-2 values?

3. Can you generalize the above?
If it help to simplify things, you can choose a specific value for N, say N = 3 or 5.
##### SOLUTION:
1. Assume N = 3 and we maximize, then for J3 = 3 to be observed as the maximum value it must be true that w3 > w2 + 1 and w3 > w1 + 2 where w3, w2 and w1 are i.i.d. N(0,1), i.e.,

2. For this probability, we need w3 > w1 + 2 or w3 > w1 + 2 or w2 > w1 + 1 or w2 > w3 - 1. Each of these four events have probabilities that can be calculated.

3. While these approach can be generalized, it is much faster to get this probability by direct Monte Carlo simulation.