62 Frequently Asked Questions In Quantitative Finance
number of taxis in the city from that event is a question
of assumptions and statistical methodology.
For this problem the obvious assumptions to make are:
1.Taxi numbers are strictly positive integers
2.Numbering starts at one
3.No number is repeated
4.No number is skipped
We will look at the probability of getting into taxi num-
ber 20,922 when there areNtaxis in the city. This
couldn’t be simpler, the probability of getting into any
specific taxi is
1
N
.
WhichNmaximizes the probability of getting into taxi
number 20,922? The answer is
N=20, 922.
This example explains the concept of MLE:Choose
parameters that maximize the probability of the outcome
actually happening.
Another example, more closely related to problems
in quantitative finance is the hat example above. You
have three hats containing normally distributed random
numbers. One hat’s numbers have mean of zero and
standard deviation 0.1. This is hat A. Another hat’s
numbers have mean of zero and standard deviation 1.
This is hat B. The final hat’s numbers have mean of zero
and standard deviation 10. This is hat C.
You pick a number out of one hat, it is−2.6. Which hat
do you think it came from?
The ‘probability’ of picking the number−2.6 from hat
A (having a mean of zero and a standard deviation of
