SOLUTION: A random variable X has the pdf x^2 if 0 < x ≤ 1 f(x) = 2/3 if 1 < x ≤ 2 0 otherwise (a)Find the median of X. (b)Sketch the graph of the CDF a

That's the graph.  Now we'll shade the area under it:



Now we find the area between the graph and the x-axis, take half
of it and find the value of x which divides the entire area into
two parts.

First we find the area under the curved (parabola) part



The right part of the graph is just a rectangle with base 1 and height , 
so its area is just .

The entire shaded area is 

So we need to find the value of x such that half the area  is left of
it and the other half , is on the right of it.

The area under the parabola is  so that leaves  the we must take 
of the rectangle. To get the base of a rectangle with area  that has height , 
we divide the area by the height and get .  So the
median is  on a unit past 1, or .

So the median is . To indicate the median, we can draw a green
vertical line through the graph at : 



Edwin