Computing the integral of g on 1 1 we get z 1 1 g x

Computing the integral of g on ( -1 , 1 ) we get
Z
1
-1
g ( x ) dx =
Z
b
- b
cos( x ) dx = 2 sin( b ) .
There is one solution for 2 sin( b ) = 1 for b 2 [0 , ⇡ / 2], it is b = arcsin(1 / 2) = ⇡ / 6. For
this choose of b the function g is a pdf.
3.9 Suppose that the random variable X has cumulative distribution function given by
F ( x ) =
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
0 , if x < 1 ,
1 / 3 , if 1  x <
3
2
,
1 / 2 , if
3
2
 x <
5
3
,
3 / 4 , if
5
3
 x <
9
5
,
1 , if x ≥
9
5
.
(a) Find the range of X .
(b) Show that X is discrete, and find its probability mass function.

Solution:
3.13 Choose a point uniformly at random in a unit square (square of side length 1). Let X
be the distance from the point chosen to the nearest edge of the square.
(a) Find the cumulative distribution function of X .
(b) Find the probability density function of X .