A question about a joint pdf and how to work out probabilities?

[laura_a]

Homework Statement

Let X1, X2, X3,...Xn be independent and indentically distributed with pdf

f_X(x)= 4x^3 for 0 < x < 1

First I had to calculate the joint pdf of distribution of X1, X2, X3..Xn which I did
I got

4(X1*X2*X3*...*Xn)^3

Which I'm hoping is correct.

a) determine the probability that the first observation X1
is less than 0.5
b) determine the probability that ALL observations are less than 0.5.
c) use (b) to deduce then that the probability that the largest observation is less than 0.5

Homework Equations

I've worked out the joint pdf which is 4(X1*X2*X3*...*Xn)^3

But I'm not sure what value I can plug into find out the probability because there are so many X's? Can anyone point me in the right direction? THANKS :)

 

[zhentil]

You don't need to calculate the joint pdf, but even if you did, your answer is wrong (you would need 4^n out front). Here's a hint in the right direction: I certainly hope you can do (a). For (b), what is the probability of tossing 8 heads in a row? This should give you a hint as to what you should do to the answer in (a). For (c), if the largest is less than 0.5, what does that mean about the rest of them? Use (b) now.