Finding the Joint and Density Functions for Independent Uniform Random Variables

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Homework Statement


Let X and Y be independent uniform (0,1) random variables.

a. find th ejoint density of U=X, V=X+Y.

b. compute the density funciton of V.

Homework Equations





The Attempt at a Solution



Part a. is not a problem. I don't understand how the bounds for part b. are set up. The book says: for 0<V<1, fv(v)=\int_{o}^{v}du
and for 1\leqv\leq 2: \int_{v-1}^{1}du

Could someone explain this to me?
 
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Could somebody explain this in English?
 

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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