Math question.

[Robin.]

Does anyone know how to find the Mode of the Probability Distribution Function that follows:

f(x) = 12x²(1-x)

I worked out that if I say:

12x² - 12x³ = 1

and then solve for x, then x should be the mode
but I'm not sure how to solve for x for one.
But secondely that seems to hard, there has to be a simpler way...

[Janet_Girl]

Would not the answer Be f = 12x-12x²

[noxdraconis]

Assuming that this is a normal distribution, the mode should be 2/3.  The mean, median, and mode of a normal distribution occur at the function's local maximum.  By taking the derivative of your funtion and setting it equal to zero, the local extrema are found to be at x=0 and x=2/3.  By doing a second derivative test on both points, it turns out that x=0 is a minimum and x=2/3 is a maximum (and thus the answer).

[old1]

This is not a 'normal distribution'  as it is not gaussian.  Wouldn't the mode be where the derivative is zero and the magnitude of the second derivative is smaller? x=0

I believe the mode is the value that occurs most frequently.  The 'normal distribution' has only one peak and is monotonic around that.  As mentioned, the mean, mode, and median occur at the peak.

Some other distribution may not spend most of the time at the peak.

[Robin.]

Assuming that this is a normal distribution, the mode should be 2/3.  The mean, median, and mode of a normal distribution occur at the function's local maximum.  By taking the derivative of your funtion and setting it equal to zero, the local extrema are found to be at x=0 and x=2/3.  By doing a second derivative test on both points, it turns out that x=0 is a minimum and x=2/3 is a maximum (and thus the answer).

Thanks that helps alot. The answer page said the mode was 2/3 so your right, I just needed to know how they got that answer.

I feel silly now, I was thinking that 1 was the highest possible value of the equation, thus the max, and so I had to find the value x that "maximized" the equation. My book wasn't to helpful in that area...
Post Merge: November 28, 2009, 09:01:42 AM

By doing a second derivative test on both points, it turns out that x=0 is a minimum and x=2/3 is a maximum (and thus the answer).

One more question.
How does the second derivitive test work?
Do you plug in 0 and 2/3 into the second derivitive? or do you solve for x again...and then what?
Post Merge: November 28, 2009, 10:02:07 AM
.....Thanks for your help! 
Post Merge: November 28, 2009, 02:20:04 PM
.... i have another question...

If I have the PDF f(x,y) = 2(x+y)
and i know that:
The PDF of X is g(x) = 1 + 2x - 3x²
and
The PDF of Y is h(y) = 3y²

How do I tell if X and Y are Independent or not?
Post Merge: November 28, 2009, 03:23:51 PM
If it helps... I'm on problem #9 on this webpage: http://www.math.uah.edu/stat/dist/Joint.xhtml

You have to use firefox to view it.
And the little questionmark boxes give the answeres, just not how the answers wer got.