Find Density Functions of X, Y, Z Variates

iHeartof12
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The random variable X assumes the values 1,2,3 and 4 with equal probability. Find the density functions of the following variates:

Attempted solutions:

X 1 2 3 4
Pr(X) 1/4 1/4 1/4 1/4

a) Y=1-2X

Y -1 -3 -5 -7
Pr(Y) 1/4 1/4 1/4 1/4

b) Z= X/(X+1)

Z 1/2 2/3 3/4 4/5
Pr(Z) 1/4 1/4 1/4 1/4

Are my solutions for parts a and b correct?
 
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Yes, this is correct.
 
Thank you. How'd your exam go today? I'm ready to get my exam over tomorrow I have a few more next week that I'm also studying for. Ahh the life of a college student. lol.
 
It went fine except for one problem that made absolutely no sense to me. Don't forget to study the pdf of a random variable of a decreasing function, not just an increasing one. There was no extra credit :( But I guess that's what you get when you can do six and he'll grade your best five.
 

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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