Page 74 - DMTH404_STATISTICS

Statistics

Notes Let us consider a special case of the above problem when X , ...., X are independent and identically
1 n
distributed with uniform distribution on [0, I]. Then

 0 if x  0

x
F(x) = x if 0   1

 1 if x  1

and

 1 if 0  x 1
f(x) = 
 0 otherwise
In this case

1-w
1
n-2
 (w ) = n(n – 1)  w dw , if 0 < w < 1
1
l
2
w 1 1
0
n-2
= n(n – l) w (1 w ) 1 , if 0 < w < 1
1
1
= 0 , otherwise
Now for a short exercise
In the next three sections we shall discuss three standard distributions each of which appear as
the distribution of a certain function of standard normal variable. We shall make use of the
different approaches discussed in this unit to obtain their distribution functions. All these
distributions play an important role in statistical inference which will be discussed in Block 4.
5.3 Summary

 Let (X,Y) be a random vector with the joint density function fx, y (x,y). Let g(x,y) be a real-
valued function defined on R x R. For z E 2, define
D = {(x, Y) : g(x, y)  z}
z
Then the distribution function of Z is defined as

P [Z z] =   f X,Y (X,Y) dx dy

D z
Theoretically it is not difficult to write down the distribution function using (1). But in
actual practise it is sometimes difficult to evaluate the double integral.
 Suppose (X , X ) is a bivariate random vector with the density function fx , x (x , x ) and we
1 2 1 2 l 2
would like to determine the distribution function of the density function of Z = g (X , X ).
1 1 1 2
To determine this, let us suppose that we can find another function Z = g (X , X ) such that
2 2 1 2
the transformation from (X , X ) to (Z , Z ) is one-to-one. In other words to every point (x ,
1 2 1 2 1
x ) in R , there corresponds a point (x , x ) in R given by the above transformation and
2
2
2 1 2
conversely to every point (z , z ) there corresponds a unique point (x , x ) such that
1 2 l 2
z = g (x , x )
1 1 1 2
z = g (x , x )
2 2 1 2

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