continuous random variable question

[Guest]

if f(x) = 348x - x^2 - 29700/18432 for 150<=x<=198
or 0 elsewhere

How can i show that f(x) is a probability density function.

What would be a suitable empirical test to evaluate the above estimated model?

 

[pka]

Show these two facts:
\(\displaystyle \L
\begin{array}{l}
x \in [150,198] \Rightarrow f(x) \ge 0 \\
\int\limits_{150}^{198} {f(x)dx} = 1 \\
\end{array}.\)

 

[marcmtlca]

because pdfs are such that:

1) they are nonnegative everywhere

2) If you integrate the function over the support then the integral is 1.

The reason for this is that the pdf must obey the laws of probability. The important ones here are that the probability of an event occurring can never be negative. And that if you sum up the probabilities of everything (disjoint) that can happen in your space you get 1.