Distribtution of the maximum of three uniform random variables. How do I get the cumulative density function of $Y$? $X$ is a continuous random variable with pdf
$$f(x) = 1,\quad   0 < x < 1. $$
Three independent observations of $X$ are made. Find the pdf of $Y$ where $$Y = \max\{X_1,X_2,X_3\}.$$

The pdf or cdf you are looking for is part of what is called the "order statistics": https://en.wikipedia.org/wiki/Order_statistic
The easiest path is through the cdf, using the following transformations:
$F_Y(y) \ = \ P(Y\leq y) \ = P(\max{\{X_1,X_2,X_3\}} \leq y)$
$F_Y(y) \ = \ P(X_1 \leq y \ \& \ X_1 \leq y \ \& \ X_1 \leq y).$
Using independence of the $X_k$:
$F_Y(y) \ = \ P(X_1 \leq y) P(X_2 \leq y) P(X_3 \leq y)=(y)^3$
It suffices then to derive wrt $y$ to obtain the pdf:
$$f_Y(y)=3~y^2$$
Edit : Very sorry : I made a mistake that I have corrected now (confusion between $\min$ and $\max$)
