2011 T1.a.2

[JalilaZ]

Hi David,

I was going through the spread sheet and found in the first tab of the aforementioned excel file the PDF for IBM having a probability of more than 1. I tried it myself on a seperate excel file and got the same result.so if we want to find the zero return probability we will get 1.2 which is impossible.

Wondering what's the logic ?

Thanks
Jay

 

[David Harper CFA FRM]

Hi Jay,

Good observation. f(x) is just the height of the function for a given x, it is not quite the probability. In a continuous distribution (e.g., normal), there are not exact local probabilities but rather intervals: a <= X < = b. Just as the total area under the curve is equal to 1.0 (100%), the probability for an interval is the AREA of the interval under the curve, not just the "height" of the function.

So the probability PR [a <= X <= b] = the integral of f(x) over the interval from a to b, which for a narrow interval we can approximate with Pr [a <= X <= b] = dx * f(x); see continuous at http://en.wikipedia.org/wiki/Probability_distribution

In this way the probability is not f(x) but rather dx * f(x), which in the case of a small interval looks like a tall, skinny rectangle. Even as the "height" can be greater then 1.0, the product of the height and the width will not exceed 1.0. Or, as the interval width increases and a rectangular becomes too crude to describe the area, the area under the curve can never be greater than the whole area of 1.0, even as the f(x) may breach 1.0. Hope that helps!

David