Hello again everyone,

I'm having some trouble understanding an example for a probability density function of a cont. RV.

The part I'm not getting is the integrals. I need an elaboration on what is going on with the calculations

so I can see what is happening. I assume that it has something to do with derivatives but I have never

seen or don't remember what the $\displaystyle \mid^2_{-1}$ etc.. or what the $\displaystyle \int$ symbol means.

Thanks for your help.

Here is the example:

[Question]

Suppose that the error in the reaction temp., in Celsius, for a controlled lab. experiment is a cont.

RV "X" having the pdf:

$\displaystyle f(x)=\left\{\begin{array}{cc}\frac{x^2}{3},&-1 < x < 2\\0, & \mbox{ elsewhere }\end{array}\right.$

- Verify that $\displaystyle f(x)$ is a density function
- Find $\displaystyle P(0 < X \leq 1)$

[SOLN]

- $\displaystyle f(x) \geq 0$, by def.

$\displaystyle \int^\infty_{-\infty}f(x)dx = \int^2_{-1} \frac{x^2}{3}dx = \frac{x^3}{9}\mid^2_{-1} = \frac{8}{9} + \frac{1}{9} = 1$

- $\displaystyle P(0 < X \leq 1) = \int^1_{0}\frac{x^2}{3}dx = \frac{x^3}{9}\mid^1_{0} = \frac{1}{9}$