1. Consider the random variable X with cumulative distribution function F(x) = (1/2) + (1/pi)tan^-1
(x), -infinity < x < infinity.
Find the 25%, 50%, and 75% quantiles of this distribution
Solution
As F(x) = 1/2 + (1/pi) tan^-1(x)
Let y = F(x) = 1/2 + (1/pi)tan^-1(x)
Then, (y - 1/2) = (1/pi) tan^-1(x), so
pi(y-1/2) = tan^-1(x), so
tan(pi(y-1/2) ) = x
Thus, the 25th percentile is
tan(pi(1/4-1/2) ) = x
tan(-pi/4) = x
x = -1
The 50th percentile is
tan(pi(1/2-1/2) ) = x
tan(0) = x
x = 0
The 75th percentile is
tan(pi(3/4-1/2) ) = x
tan(pi/4) = x
x = 1