I had this problem which I had no idea how to solve.
Random variable X has a cumulative distribution function with the density: $$f(x)=\left\{ \begin{array}{l} c(1+x)&x∈[0,1] \\ 0&x∉[0,1] \end{array} \right.$$
A) Define the unknown constant c and the distribution function of X
B) Define the Expected value of the random variable $1+\frac{X^2}{2}$
C) Random variable Z is given by $Z=\sqrt X -1$ find the distribution function of Z
For A) I understand that to calculate the constant c we need to integrate: $$1=\int f(x)dx = \int_0^1 c(1+x)dx = c[x]_0^1+c[\frac{x^2}{2}]_0^1=1c+\frac{1}{2}c=\frac{3}{2}c => c=2/3$$ And for the distribution function: $$D(x)=P(X<=x)=\int_{-∞}^xf(t)dt = \int_0^x\frac{2}{3}(1+t)dt=...=\frac{2}{3}(x+\frac{x^2}{2})$$ And afterwards I'm lost. Not even sure on the beforehand things.