1. parabolic probability density function

Hi, I'm just in an introductory Stats class,
learning about Normal, gamma, F, t, W, Chi-squared distributions, etc, binomial series, joint density functions, mgf and confidence intervals, etc.

There's this question:

You wish to fence off a rectangular paddock on one side of a river running through your property in a straight line. No fence is required alongside the paddock formed by the river. The fence you will use is rolled up in a shed, and you are at the moment not quite sure how long it is. However, you are certain that it is between 3km and 5km long, and your certainty regarding its length can be represented by a parabolic probability density function which tapers off to zero at 3km and 5km.

Find and sketch the pdf of Y, the area of the largest paddock you will be able to fence off.

Find EY, mode, and median of Y.

The problme here is that we do not know what kind of distribution function it is. If it's just a simple parabola... it would be a High School Maths. Please advise me on what kind of distribution that parabolic probability refers to.

2. Originally Posted by zangbangapda
Hi, I'm just in an introductory Stats class,
learning about Normal, gamma, F, t, W, Chi-squared distributions, etc, binomial series, joint density functions, mgf and confidence intervals, etc.

There's this question:

You wish to fence off a rectangular paddock on one side of a river running through your property in a straight line. No fence is required alongside the paddock formed by the river. The fence you will use is rolled up in a shed, and you are at the moment not quite sure how long it is. However, you are certain that it is between 3km and 5km long, and your certainty regarding its length can be represented by a parabolic probability density function which tapers off to zero at 3km and 5km.

Find and sketch the pdf of Y, the area of the largest paddock you will be able to fence off.

Find EY, mode, and median of Y.

The problme here is that we do not know what kind of distribution function it is. If it's just a simple parabola... it would be a High School Maths. Please advise me on what kind of distribution that parabolic probability refers to.
Let the pdf of Y be f(y). You require f(3) = f(5) = 0.

Therefore f(y) = a(y - 3)(y - 5).

You require $\int_3^5 f(y) \, dy = 1 \Rightarrow a \int_3^5 (y-3)(y-5) \, dy = 1$.

Integrate and solve for a.

Once you have the pdf it is simple to calculate E(Y), mode and median.