Basic Expectation

[RoX@r]

hi guys,

this problem looks simple but tricky. give it a try.

Q. Suppose X1 and X2 are real-valued random variables with f as their common p.d.f. Suppose (x1, x2) is a sample generated by these random variables. The expectation of the number of observations in the sample that fall within a specified interval [a, b] is??

[Dason]

Can you figure out the probability that X1 falls in the interval [a,b]?

[RoX@r]

Basically its a multiple choice question.
These are the 4 options.
a) whole square of integration of f(x)dx with limits a to b.
b) integration from a to b of x^2f(x)dx
c) 2 integration from a to b of f(x)dx
d) integration from a to b of xf(x)dx

Though i think the right ans. is b). i am having trouble to show it.

[Dason]

What is your justification for thinking it is b?

(I don't believe it's b)

[RoX@r]

because clearly a) and c) do not follow the def. of exp. which is int. from a to b xf(x)dx as there is no x term in the int. and option d) is simply E(x). but we need E(x1, x2)!!

This is as far as i could think. correct me if i am wrong. and pls do justify your ans. if u managed to find one.

[Dason]

Take a step back.

You're looking for the expected number of your random variables that fall in an interval. What type of distribution does this response have? It could be 0, 1, or 2 right? Think about that for a little while.

[RoX@r]

well not sure what type of distribution it can follow. but now i think ans. is d). workd with an eg. If X1 is a r.v for even no.'s after tossing a die and X2 for odd. Then expected no. of obs. falling within 4 to 6 is just integration from 4 to 6 xf(x)dx. Am i in the same page?

[Dason]

well not sure what type of distribution it can follow.

You probably do. Come on - what kind of distribution models the number of successes in a finite number of trials. For example what would you use to model the number of heads observed when you flip a fair coin 3 times?

but now i think ans. is d). workd with an eg. If X1 is a r.v for even no.'s after tossing a die and X2 for odd. Then expected no. of obs. falling within 4 to 6 is just integration from 4 to 6 xf(x)dx. Am i in the same page?

No - (d) doesn't give you the conditional expectation. You need to do a little more work than that to get a conditional expectation.