Correlation and mathematical expectation question

[Barioth]

Hi, I have these 2 problem, that I'm not so sure how to handle.

1-Let $$X_1,X_2,...,X_n$$ independent Random variable that all follow a continuous uniform distribution in (0,1)
a) Find $$E[Max(X_1,X_2,...,X_n)]$$
b) Find $$E[Min(X_1,X_2,...,X_n)]$$

where E is for the mathematical expectation. I'm not so sure how to tackle such a question.

2-Let$$ X_1, X_2, X_3 and X_4$$ are Random variable with no correlation two by two.
Each with mathematical expectation = 0 and variance =1. Evaluate the Correlation for

a-$$ X_1+X_2 and X_2+X_3$$

b-$$X_1+X_2 and X_3+X_4$$

I know that $$Corr(X_1+X_2,X_2+X_3)=\frac{Cov(X_1+X_2,X_2+X_3)}{ \sqrt {Var(X_1+X_2)*Var(X_2+X_3)}}$$

All I can think of is using the CTL, but since I don't know if they're independent I can't use it? Also we've seen the CTL after been giving this problem.

Thanks for passing by!

 

[chisigma]

Re: correlation and mathematical expectation question

Hi, I have these 2 problem, that I'm not so sure how to handle.

1-Let $$X_1,X_2,...,X_n$$ independent Random variable that all follow a continuous uniform distribution in (0,1)

a) Find $$E[Max(X_1,X_2,...,X_n)]$$

b) Find $$E[Min(X_1,X_2,...,X_n)]$$

In...

http://www.mathhelpboards.com/f52/unsolved-statistic-questions-other-sites-part-ii-1566/index6.html

... it has been found that...

$\displaystyle E \{ Max X_{i}, i=1,2,...,n\} = \frac{n}{n+1}$ (1)

It is very easy to find that is...

$\displaystyle E \{ Min X_{i}, i=1,2,...,n\} = \frac{1}{n+1}$ (2)

Kind regards

$\chi$ $\sigma$

 

[Barioth]

Re: correlation and mathematical expectation question

Thanks it's a pretty clean solution!

 

[Barioth]

Re: correlation and mathematical expectation question

I was wondering, if I'm given the probability density of X and Y and we do not know if they are inderpendant. Is there a way to find $$f_{X,Y}(x,y)$$?