Finding distribution function of a variable

Question

Let's say we have a random variable $X$ with uniform distribution from $0$ to $1$, so $X \in U(0,1)$. How can we find a distribution function of some other variable that can be expressed using X, for example: $$Y=\frac{1}{X+1}$$ What is a general procedure of finding distribution functions like that of $Y$?

Answer 1

Since $x$ gets from $0$ to $1$, $y$ gets from $1/2$ to $1$. You know that X is uniformly distributed from $0$ to $1$ so you know the functions $f_X(x)$ and $F_X(x)$ Now: $$F_Y(y) = P(Y< y) = P(1/(X+1) < y) = P(X > 1/y - 1) = 1 -P(X < 1/y - 1)=\\= 1 - F_X(1/y - 1)$$ and replace it to $F_X(x)$