Is 3/256(x-2)(10-x) for the intergral [2,10] a probablity density function?

Question

Is it? Doesn't the answer come out negative? This is a problem from our calculus textbook that looks a bit fishy to me. It then asks you to do the following, assuming it is a probability density function (which I'm sure it isn't)...

A) Determine the expected value and standard deviation

b) Determine the median

c) find the probability that x is within one standard deviation of the mean.

Answer 1

The function can very well be a probability distribution. The expected value is given by

m = ∫x*P(x)dx [limits 2 - 10] (this comes out m = 6)

the standard deviation is

s =√[ ∫(x-m)^2*P(x)dx] [limits 2 - 10] (this comes out s = 1.8)

Since the distribution is symmetric, the median is the same as the mean (expected value): half of the area is above and half below the expected value.

The probability that x is within one std dev is

∫P(x)dx [ limits m-s and m+s)] this comes out .63

You can see the work here (integrated numerically by MathCad)
http://img299.imageshack.us/img299/5619/probdistdd0.png

You wll probably want to work the integrals by multiplying out the integrands and integrating explicitly, then evaluating at limits.

Answer 2

I'm a bit tired and haven't done PDF's in a while.
THe defined integral is 1 though. So as far as what i remmeber, it is a PDF.

The rest i can't really help you out with. I'm at a point where i just use a computer program and stick things in .