Download is a random variable

Probability
Theory
School of Mathematical Science
and
Computing Technology in CSU
Course groups of Probability and Statistics
§2.3distribution line of random variable
function
一、importing question
二、distribution of disperse random variable
function
一、 importing question
there are seven red ball，three white ball in the bag.now take two
ball from any of the bag,if you got one red ballyou have ten yuan,
found out what is the probability you have twenty yuan?
If  is the number you got red ball,so ＝10 is
random variable
for settling the similar question,
fellowing we discuss distribution of disperse
random variable function
二、distribution of disperse
random variable function
If y = g( x) is the single value function of x，
 is a random variable , and ＝g ( )
Is a random variable too ,what more when

is x ＝g ( x)
For exsample 1. when import somegoods n pieces，
everyone is a yuan。By the compact，
if there is one reject in the n pieces，
the exiter have to repay 2a yuan。now，
the reject of the n pieces (  )is a random variable ，
and repay of exiter   2a is a random variable too.
if the Probability of every good to be reject is p
because of P(  k )
k
k
nk
C
p
(1

p
)
,0  k  n
= n
but (  2ak )  (  k ),0  k  n
So
P(  2ak )  C p (1  p)
k
n
k
nk
,0  k  n
if r.v. the distributing rule of  is
P(  ai )  pi ,
i  1,2,
from the function f(x) is the single value function
about the real variable,when  is finity value a i,
i  1, 2...We can get r.v. all the probability value
b
i
i  1, 2... so The probability distributing of  is:
P(  b j ) 

k : g ( ai ) b j
P(  ai ), j  1, 2,
Example 1:the probability
distributin of x is
X
-1
0
pk
1
8
1
8
1
2
1
4
1
2
Found out the distributing rule of Y 1= 2X – 1
and Y 2= X 2
Y1
-3
-1
pi
1
8
1
8
1
1
4
3
1
2
Y2
1
0
pi
1
8
1
8
1
4
Y2
0
1
4
pi
1
8
3
8
1
1
2
4
1
2
Example 2: if  is the random variable
of poisson distribution that which
parameter is 
1, x is even



f ( x)  0，x=0

1, x is odd number


Found out the distribution of   f ( )
we know that the value of is 1,0，－1

so P(  1）＝ P(  2k )
k 1



2k
 (2k )! e

k 1
P(  0)  P(  0)  e


P(  1)   P(  2k  1)
k 0



2 k 1
 (2k
k 0
e
 1)!

The distribution of the function of
planar disperse variable
If the p variable( , )
If the planer random variable( , )
we know that f(x,y) is the single function about
real variable x and y,so  =f( , ) stile is a disperse
random variable,when( , )is finity value(a j,b k )
j,k=1,2... we can found out r.v. all the posible values
of 
c i  1, 2...,so the probality distribution of
i,
is:
P(  ci ) 

k : g ( ai ) b j
P(  a j ,  bk ), i  1, 2,
Example 3: if the distributions of two abslute
Random variables x and y
X
Y
1
3
2
4
PX
0.3
0.7
PY
0.6
0.4
Found out the distribution rule of the random
variableZ=X
becauseX and Yone another indepandence, so
P{ X  xi ,Y  y j }  P{ X  xi }P{Y  y j },
X
1
3
Y
P
2 4
0.18 0.12
0.42 0.28
0.18
then 0.12
0.42
0.28
( X ,Y ) Z  X  Y
(1,2)
(1,4)
( 3, 2 )
( 3, 4 )
3
5
5
7
is
X
1
3
Y
Z  X Y
So
that
P
2
0.18
0.42
4
0.12
0.28
3
5
7
0.18
0.54
0.28
Example 4:if  ,  is tow absolute random variable
they obey diferent poisson distribution which
parameter is  1, 2found out the distribution of  =  
Because of the independence character
of  , 
1i  1 2 k i  2
P(  k )   P(  i,  k  i )   e .
e
(k  i)!
i 0
i 0 i !
k
k
i k i

 ( 1  2 )
1 2
e

i  0 i ! ( k  i )!
k
(1 +2）  ( 1 2 )

e
, k  0,1, 2..
k!
k
Example 5: if two independence random
Variable X, Y have the same
distribution,and the distribution of x is
X
0
1
P
0.5
0.5
found out: the distribution of z=max(x,y)
because x and y ane another independence
so p{x=i,y=j}=p{x=i}p{y=j}
So
X
0
1
Y
0
1
1 22
1 22
1 22
1 22
P{max( X ,Y )  i }
 P{ X  i ,Y  i }
 P{ X  i ,Y  i }
X
0
1
Y
0
1
2
12
1 22
1
 P{max( X ,Y )  0} P{0,0}  2 ,
2
P{max( X ,Y )  1}  P{1,0}  P{0,1}  P{1,1}
1 1 1 3
 2  2  2 2.
2 2 2 2
1
so the distributionof Z 0
1
3
Z=max(x,y)
P
4
4
2
12
1 22
Have a rest now