Download Discrete, Continuous and Mixed-type random variables

Discrete, Continuous and Mixed-type random variables


A random variable X is called a discrete random variable FX (x) is piecewise constant. Thus FX (x) is flat except at the points of jump discontinuity. If
the sample space S is discrete the random variable X defined on it is always
discrete.
X is called a continuous random variable if FX (x) is an absolutely continuous
function of x. Thus FX (x) is continuous everywhere on
and FX  ( x ) exists
everywhere except at finite or countably infinite points .
 X is called a mixed random variable if FX (x) has jump discontinuity at
countable number of points and it increases continuously at least at one
interval of values of x. For a such type RV X,
FX ( x)  pFX d ( x)  (1  p) FX c ( x)
where FX d ( x) is the distribution function of a discrete RV and
FX c ( x ) is the
distribution function of a continuous RV. Typical plots of FX ( x) for discrete,
continuous and mixed-random variables are shown in Fig below.
FX ( x)
1
x
Plot FX ( x) vs. x for a discrete random variable ( to be animated)
FX ( x)
1
x
FX ( x)
1
x
Plot FX ( x) vs. x for a continuous random variable ( to be animated)
FX ( x)
1
x
Plot FX ( x) vs. x for a mixed-type random variable ( to be animated)
Discrete Random Variables and Probability mass functions
A random variable is said to be discrete if the number of elements in the range of RX is finite or
countably infinite. Examples 1 and 2 are discrete random variables.
Assume R X to be countably finite. Let x1 , x2 , x3 ..., xN be the elements in RX . Here the mapping
X ( s) partitions S into N subsets s | X (s)  xi  , i  1, 2,...., N.
The discrete random variable in this case is completely specified by the probability mass
function (pmf) pX ( xi )  P( s | X ( s)  xi ), i  1, 2,...., N .
Clearly,



pX ( xi )  0 xi  RX and
 p X ( xi )  1
iRX
Suppose D  RX . Then
P({x  D}) 
p
xi D
X
( xi )
X ( s1 )
X ( s2 )
s1
s3
s2
X ( s3 )
s4
X ( s4 )
Figure Discrete Random Variable
(To be animated)
Example
Consider the random variable X with the distribution function
0
1

4
FX ( x )  
1
2
1

x0
0  x 1
1 x  2
x2
The plot of the FX ( x) is shown in Fig.
FX ( x)
1
1
2
1
4
0
1
2
x
The probability mass function of the random variable is given by
Value of the
random
Variable X  x
0
1
2
p X ( x)
1
4
1
4
1
2
We shall describe about some useful discrete probability mass functions in a later class.
Continous Random Variables and Probability Density Functions
For a continuous random variable X , FX (x) is continuous everywhere. Therefore,
FX ( x)  FX ( x  ) x  . This implies that
p X ( x)  P ({ X  x})
 FX ( x)  FX ( x  )
 0
Therefore, the probability mass function of a continuous RV X is zero for all x. A
continuous random variable cannot be characterized by the probability mass function. A
continuous random variable has a very important chacterisation in terms of a function
called the probability density function.
If FX (x) is differentiable, the probability density function ( pdf) of X ,
f X ( x), us defined as
f X ( x) 
denoted by
d
FX ( x )
dx
Interpretation of f X ( x)
d
FX ( x)
dx
F ( x  x)  FX ( x)
 lim X
x 0
x
P({x  X  x  x})
 lim
x 0
x
f X ( x) 
so that
P({x  X  x  x})
f X ( x)x.
Thus the probability of X lying in the some interval ( x, x  x] is determined by
f X ( x). In that sense, f X ( x) represents the concentration of probability just as the density
represents the concentration of mass.
Properties of the Probability Density Function

f X ( x)  0.
This follows from the fact that FX ( x) is a non-decreasing function
x

FX ( x) 

f X (u )du



f
X
( x)dx  1


P( x1  X  x 2 ) 
x2
f
 x1
f X ( x)
X
( x)dx
x0
x0  x0
Fig. P({x0  X  x0  x0 })
x
f X ( x0 )x0
Example Consider the random variable X with the distribution function
x0
0
FX ( x)  
 ax
1  e , a  0 x  0
The pdf of the RV is given by
0
f X ( x )    ax
e , a  0
x0
x0
Remark: Using the Dirac delta function we can define the density function for a discrete
random variables.
Consider the random variable X defined by the probability mass function (pmf)
pX ( xi )  P(s | X (s)  xi ), i  1, 2,...., N .
The distribution function FX ( x) can be written as
N
FX ( x)   p X ( xi )u ( x  xi )
i 1
where u ( x  xi ) shifted unit-step function given by
1 for x  xi
u ( x  xi )  
0 otherwise
Then the density function f X ( x) can be written in terms of the Dirac delta function as
n
f X ( x)   p X ( xi ) ( x  xi )
i 1
Example
Consider the random variable defined in Example 1 and Example 3. The distribution
function FX ( x) can be written as
1
1
1
FX ( x)  u ( x)  u ( x  1)  u ( x  2)
4
4
2
and
1
1
1
f X ( x)   ( x)   ( x  1)   ( x  2)
4
4
2
Probability Density Function of Mixed-type Random Variable
Suppose X is a mixed-type random variable with FX ( x) having jump discontinuity at
X  xi , i  1, 2,.., n. As already stated, the CDF of a mixed-type random variable X is
given by
FX ( x)  pFD  x   1  p  FC  x 
where FD  x  is the conditional distribution function of
FC  x  is the conditional distribution function given that
The corresponding pdf is given by
X
X
given
X
is discrete and
is continuous.
f X ( x)  pf D ( x)  (1  p) fC ( x)
where
n
f D ( x)   p X ( xi ) ( x  xi )
i 1
and fC  x  is a continuous pdf.
Suppose RD  {x1 , x2 ,..., xn } denotes the countable subset of points on RX such that the
RV
X
is characterized by the probability mass function pX  x  , x  SD . Similarly let
RC  RX \ RD be a continuous subset of points on RX such that RV is characterized by the
probability density function fC  x  , x  RC .
Clearly the subsets RD and RC partition the set RX . If P  RD   p , then P  RC   1  p .
Thus the probability of the event  X  x can be expressed as
P  X  x  P  RD  P  X  x | RD   P  RC  P  X  x | RC 
 pFD  x   1  p  FC  x 
 FX ( x)  pFD  x   1  p  FC  x 
Example Consider the random variable X with the distribution function
0
0.1

FX ( x)  
0.1  0.8 x
1
x0
x0
0  x 1
x 1
FX ( x)
1
The plot of FX ( x) is shown in Fig.
FX ( x)
can be expressed as
FX ( x)  0.2 FX d ( x)  0.8FX c ( x)
where
0

FX d ( x)  0.5
1

x0
0  x 1
x 1
and
0

FX c ( x)   x
1

x0
0  x 1
x 1
The pdf is given by
f X ( x)  0.2 f X d ( x)  0.8 f X c ( x)
where
0
1
x
f X d ( x)  0.5 ( x)  0.5 ( x  1)
and
0  x 1
1
f X c ( x)  
0
elsewhere
f X ( x)
1
0
1
x
Example
X is the RV representing the life time of a device with the PDF f X  x  for x>0. Define
the following random variable
yX
if X  a
a
if X  a
RD  {a}
RC   0, a 
p  P  y  D
 P  X  a
 1  FX  a 
 FX  x   pFD  x   1  p  FC  x 