Correlation and mathematical expectation question

[Barioth]

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

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

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

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

a-\(\displaystyle X_1+X_2 and X_2+X_3\)

b-\(\displaystyle X_1+X_2 and X_3+X_4\)

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

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!

 

[chisigma]

Re: correlation and mathematical expectation question

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

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

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

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

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 \(\displaystyle f_{X,Y}(x,y)\)?

 

[blue_raver22]

Hi there,

I can provide some guidance on how to approach these questions. For the first problem, we are given n independent random variables that follow a continuous uniform distribution in the range of (0,1). This means that each of these variables has an equal chance of occurring within this range. To find the expected value, we can use the formula E[X] = (a+b)/2, where a and b are the lower and upper limits of the distribution. In this case, a=0 and b=1, so E[X] = (0+1)/2 = 0.5. Since we are looking for the expected value of the maximum and minimum of these n variables, we can use the fact that the maximum value will be at least as large as any of the individual values, and the minimum value will be at most as small as any of the individual values. Therefore, for part a) we can say that E[Max(X_1,X_2,...,X_n)] = E[X_i] = 0.5, and for part b) we can say that E[Min(X_1,X_2,...,X_n)] = E[X_i] = 0.5.

For the second problem, we are asked to find the correlation between two pairs of random variables, X_1+X_2 and X_2+X_3, and X_1+X_2 and X_3+X_4. The formula for correlation is given correctly, but we also need to know the covariance between these pairs of variables. Since we are given that X_1, X_2, X_3, and X_4 have no correlation, we can assume that the covariance between any two of these variables is 0. Therefore, for both parts a) and b), we can say that Cov(X_i+X_j, X_k+X_l) = 0, and the correlation will be 0 as well.

I hope this helps in tackling these problems. Remember to always use the appropriate formulas and assumptions when solving mathematical problems. Good luck!