Probability Density Question

[diggetybo]

Hello,

I understand how taking the integral of the probability density results in the cumulative distribution, and conversely, taking the derivative of the CDF results in the PDF, but what I'm not so clear on is if the typical calculus partial derivative interpretation is meaningful in this case.

The area under the PDF curve, for practical purposes, sums to 1, that's why taking the integral works, but what does this imply about the Y axis? The typical case, is Y= .398 for the max of the function. But what does this magnitude really mean? Is it the change in probability? Something akin to what we might observe in physics? Like the relation we see between acceleration and velocity, where acceleration is the change in velocity (as acceleration is the derivative of velocity)? Or are the y magnitudes that result when plotting the derivative PDF from the CDF meaningless?

To probe into this, I plotted the first difference of my CDF, and saw the same shape as the PDF, but the magnitudes were nowhere close. The highest change in probability was .07, a far cry from .398.

If someone can enlighten me, I'd do a backflip.

Thank you

[startz]

Maybe I can help a little. First, let's assume we're talking about a continuous pdf.

Loosely speaking, the height of the pdf at one point compared to the other gives the relative probability. I say "loosely" because formally the probability at any point is zero.

The right way to think about the pdf is to imagine the probability of being less than point B minus the probability of being less than point A (B>A). That's the difference in the two cdfs, F(B)-F(A). Now think of the distance between B and A shrinking toward zero. As that happens (F(B)-F(A))/(B-A) goes to the pdf f().

[diggetybo]

Thanks startz,

Your explanation was really helpful, especially how F(b)-f(a)/(b-a) reduces to the pdf. I'm verging on an epiphany :)

Keeping in mind we cannot evaluate the pdf at a point, I'm still trying to illustrate the importance of the point in a graphical way, to show where the biggest return occurs for x. So, would including the y magnitude on the pdf be helpful? It seems nearly all pdf's have extremes at .398. I guess my question boils down to, does the corresponding Y value of interesting x values (max, diminishing retrns, increasing returns, ect) offer any useful insights? You used the term relative probability.

If it helps, I'll throw in some figures. Let's say I have a y of .398 and another at .240, is this y value magnitude information telling me anything new, that the ingreal formed by subtracting the two points to form an area isnt?

[startz]

Extreme values of a pdf can be any positive value. If you're getting .398, it's probably a coincidence. Take a look at

Code: Select all

series t = @tdist((@trend-50)/20,2)

[diggetybo]

yea, i think .398 is when it's centered at zero with normal standard deviation, or all that stuff. But yes, that does clear that question up. It must have been a coincidence.

And lastly, the above question about relative probability, or whatever you want to call it, concerning whether we get any unique information from the y values?

[startz]

I don't think you get anything from the pdf that you don't get from the change in the cdf.

[diggetybo]

Ok, just was extra curious about that part. Thanks again.

As promised, I did a backflip. Or attempted one rather.

[startz]

Ok, just was extra curious about that part. Thanks again.

As promised, I did a backflip. Or attempted one rather.

:)