# Probability density functions

1. Nov 9, 2007

### t_n_p

1. The problem statement, all variables and given/known data

3. The attempt at a solution
I know how to compute something like Pr(x<0.25) for example, but I'm unsure how to do it for an exact number like in question (ii). I attempted to integrate and then sub x=1/4 where neccisary, but to no avail!

Part (iii) I really have no idea!

Would be grateful is someone could explain these to me. Thanks!

2. Nov 9, 2007

### Staff: Mentor

ii) Is a trick question in a sense. Hint: the P(X=1/4) is the same for all continuous random variables with a continuous pdf.

iii) What is the definition of expected value?

3. Nov 9, 2007

### t_n_p

Ok, so P(x=1/4)=0 for all condinuous random variables with a continuous pdf, how come?

Re part (iii) straight swap (cos(pi*x)) for X into this equation?

If I do that, i get a nasty integral of cos(pi*x)*sin(pi*x)dx

Pardon my silly questions, I'm rusty as hell..

Last edited: Nov 9, 2007
4. Nov 9, 2007

### Staff: Mentor

For any continuous function f, what is

$$\lim_{\epsilon\to 0}\int_{a-\epsilon}^{a+\epsilon}f(x)dx$$

Re part (iii). This is a simple integral. If the factor of pi is throwing you off, think of this as

$$\int \cos(ax)\sin(ax)dx$$

and then set $a=\pi$ after integrating.

5. Nov 9, 2007

### t_n_p

Ah ha, got you for part (ii), it's not just 1/4 then, it's any number correct!?

re (iii), i'm still bloody lost. tried integration by parts, but the integration part just returns the same integral so it's like going around in a circle (if you know what i mean). I can't think of any other way to solve it

6. Nov 9, 2007

### Staff: Mentor

Try a simple u-substitution.

7. Nov 10, 2007

### HallsofIvy

Staff Emeritus
Let u= sin($\pi$x).

8. Nov 10, 2007

Ah gotcha!