# Conditional expectation

Question 1)

I have X and Y independent stoch. variables

What is E[X^2 * Y | X] ?

does it generally hold that if X and Y are independent, then every function of X (eg X^2) is independent of Y?

Does E[X^2 * Y | X] then become E[X^2|X]*E[Y|X] = E[X^2|X]*E[Y] since X^2 is independent of Y?

I'm stuck here... (If the above is correct, what is E[X^2|X)?

Question 2)

B(t) is Brownian motion (starting at 0), and 0<t1<t2<t3<t4

E[B(t4)-B(t2) | B(t3)-B(t1)] = ?

I'm stuck here because I don't know what the expression evaluates to when the intervals overlaps. E[B(t4)-B(t3) | B(t2)-B(t1)] = E[B(t4)-B(t3)] = 0 since the intervals are disjoint...? But what about the above expression?

Thank you