In the middle of a complicated dance routine, five dancers are located across the width of the stage. Let X1,..., X5 denote their positions across the stage. Assume that the Xj's are independent and that each Xj is uniformly distributed on [0,100], where "0" denotes the far left side of the stages, and "100" denotes the far right side of the stage, and the measurements are given in feet.
a. Find the density of the person located closest to the left-hand side of the stage, i.e., find fY(y) for Y = min(X1,..., X5). (Y is just the 1st order statistic, i.e., Y = X(1))
b. Find the probability that nobody is located within 20 feet of the left-hand side of the stage.
c. Find the probability that at least two people are located within 30 feet of the left-hand side of the stage.