Conditional expectation

[sony]

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

 

[mighty2000]

for your question. Let me address each of your questions separately.

Question 1:

To calculate E[X^2 * Y | X], we can use the definition of conditional expectation:

E[X^2 * Y | X] = ∫x∫y (x^2 * y) * f(x,y) dy dx

Where f(x,y) is the joint probability density function of X and Y.

To answer your second question, if X and Y are independent, then every function of X is independent of Y. This is because the definition of independence states that the joint probability distribution of two random variables is equal to the product of their individual probability distributions. Therefore, if X and Y are independent, then E[X^2 * Y | X] becomes E[X^2 | X] * E[Y | X] = E[X^2 | X] * E[Y].

E[X^2 | X] is the conditional expectation of X^2 given X. This can be calculated using the definition of conditional expectation:

E[X^2 | X] = ∫x (x^2 * f(x)) dx

Question 2:

To calculate E[B(t4)-B(t2) | B(t3)-B(t1)], we can use the definition of conditional expectation:

E[B(t4)-B(t2) | B(t3)-B(t1)] = ∫x (x * f(x)) dx

Where f(x) is the probability density function of B(t4)-B(t2) given B(t3)-B(t1). This can be calculated using the properties of Brownian motion.

You are correct that E[B(t4)-B(t3) | B(t2)-B(t1)] = E[B(t4)-B(t3)] = 0, since the intervals are disjoint. However, the expression E[B(t4)-B(t2) | B(t3)-B(t1)] is not equal to E[B(t4)-B(t3)], as the intervals overlap and the conditional expectation takes into account this overlap.

I hope this helps clarify your questions. Please let me know if you have any further questions.