
Whereas for a discrete random variable we
sum probabilities up to a point x to obtain the
cumulative distribution function P  X  x  , in
the continuous case we integrate.

The CDF of a continuous random variable
with density function f(x) is
x
F ( x)  P( X  x) 

f (t )dt


The function F(x) takes values in [0,1], i.e.
0  F  x   1, and is non-decreasing.
3
For X with pdf,
2 x if 0  x  1
f ( x)  
0 otherwise
the CDF is given by
0 if x  0
 2
F ( x)   x if 0  x  1
1 if x  1

4


For this example,
0
x
x
 1
F ( x)   
dt 
360
 0 360
1
x  0,
0  x  360,
x  360.
In general, for a uniform rv on [A,B],
0
 x  A
F  x  
B  A
 1
x  A,
A  x  B,
xB
5

Let X be a continuous random variable with
pdf f(x) and cdf F(X). Then for any number a,
P  X  a   1  F (a)
and for any two numbers a and b,
P  a  X  b   F b   F  a 
6

If X is a continuous rv with pdf f(x) and cdf F(x),
then at every x at which the
derivative F   x  exists, F   x   f ( x) .
For
0 if x  0
 2
F ( x)   x if 0  x  1
1 if x  1

,
2 x if 0  x  1,
f ( x)  
0 otherwise.

Let 0<p<1. The (100p)th percentile of a rv,
denoted by   p  , is defined by
p  F   p   
  p

f ( y )dy


The median is the 50th percentile. The text
denotes the median by  .

A continuous distribution whose pdf is
symmetric has median  equal to the point
of symmetry.

Expected values of continuous random
variables are computed in a manner similar to
the discrete case, with integrals replacing
sums.
10

The expected value or mean of a continuous
rv X with pdf f(x) is computed as

  EX  
 xf  x dx


For a function h(X),
E  h  X   

 h  x  f  x dx

11

The variance of a continuous random variable
with pdf f(x) and mean value  is
 V X   
2
X


x  


2
2

f  x  dx  E  X    


The variance may also be computed using
V X   EX
2
   E  X 
2
12