Correlation and mathematical expectation question

AI Thread Summary
The discussion revolves around two main problems involving independent random variables and their mathematical expectations. For the first problem, it is established that the expected value of the maximum of n independent uniform random variables on (0,1) is E[Max(X_1,...,X_n)] = n/(n+1), while the expected value of the minimum is E[Min(X_1,...,X_n)] = 1/(n+1). The second problem involves evaluating the correlation between sums of uncorrelated random variables, specifically Corr(X_1+X_2, X_2+X_3) and Corr(X_1+X_2, X_3+X_4). The discussion highlights the use of covariance and variance in calculating these correlations, although uncertainty about independence complicates the approach. Overall, the thread provides insights into mathematical expectations and correlation calculations for random variables.
Barioth
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Hi, I have these 2 problem, that I'm not so sure how to handle.

1-Let $$X_1,X_2,...,X_n$$ independant 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 independant I can't use it? Also we've seen the CTL after been giving this problem.

Thanks for passing by!
 
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Re: correlation and mathematical expectation question

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

1-Let $$X_1,X_2,...,X_n$$ independant 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

$\chi$ $\sigma$
 
Re: correlation and mathematical expectation question

Thanks it's a pretty clean solution!
 
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 $$f_{X,Y}(x,y)$$?
 
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