A random variable U follows the uniform distribution of (-1,1). Find the cumulative distribution functions and density for the transformed variables listed below.
a. $X = |U|$
b. $Y = -ln(|U|)$
c. $W = (1 – |U|)^3$
d. $T = (1 – |U|)^{1/2}$
I have a test tomorrow and really would like to understand the practice exam question.
Best Answer
You have to figure out $F_X(x)$, the CDF of $X$ (similarly for $Y$, etc.) for all real numbers $x$. One simple calculation is to find the maximum possible value $x_{\max}$ of $X$ and the minimum possible value $x_{\min}$ of $X$ and set $F_X(x) = 1$ for all $x \geq x_{\max}$ and $F_x(x) = 0$ for all $x < x_{\min}$.
Next, choose your favorite real number $x \in (x_{\min}, x_{\max})$ and write $$F_X(x) = P\{X\leq x\} = P\{U \in A\}$$ where $A$ is a set of real numbers that you need to figure out all by yourself. Remember that $A$ has the property that $X \leq x$ exactly when $U \in A$, and of course, $A$ will depend on the choice of $x$. Now, compute $P\{U \in A\}$ using the known pdf of $U$ and your knowledge of the set $A$. Integration might be required for this.
Repeat for your next favorite real number, and then the next most favorite, and so on. After a while, you might have an "Aha!" moment where you realize that for all real numbers $x$ in some interval $(\alpha, \beta)$, you will find that $F_X(x) = \gamma(x)$ for some function $\gamma(\cdot)$. Now pick a number in $(x_{\min}, x_{\max}) - (\alpha,\beta)$ and keep going. You will thus come up with a complete description of the function $F_X(x)$. Warning: It is only in very rare cases that $F_X(x)$ can be expressed by a single "formula" valid for all $x$.
Now, differentiate $F_X(x)$ to find the density $f_X(x)$.
Repeat all this for the other variables.
Trust me: it gets easier with practice. But you got to do it yourself, and struggle with finding sets $A$ and checking to make sure you have accounted for all $x$, etc; just blindly copying down what your instructor or TA writes on the blackboard or the leader of your "study group" writes on his homework solutions will not work. Learning probability theory (indeed, learning any branch of mathematics) is not a spectator sport; you have to struggle with it yourself.