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