How Do You Calculate Covariance and Correlation for X ~ U[0,1] and Y ~ U[0,X]?

asept
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I'm stuck on this problem:

Let X be uniform[0,1] and Y be uniform[0,X]. Calculate the covariance and correlation between X and Y.


thanks
 
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Just apply the definition of covariance and correlation. What are the formulae for both?
 
Cov(X,Y) = E(XY) - E(X)E(Y)
 
Yes correct, and now just apply the formulae for E(X), E(Y) and E(XY). What formulae should you use to evaluate each?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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