How to Calculate Probability Density Functions for Exact Numbers

[t_n_p]

Homework Statement

http://img204.imageshack.us/img204/2097/34629164kd2.jpg

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!

 

[D H]

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?

 

[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?
http://img233.imageshack.us/img233/5191/92698641bo2.jpg
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..

 

[D H]

For any continuous function f, what is

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

 

[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

 

[D H]

Try a simple u-substitution.

 

[HallsofIvy]

Let u= sin(\pix).

 

[t_n_p]

Ah gotcha!