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Unformatted text preview: ma We further know,
f A a forallu fU (u )
du Using this equation, we get the density function as, a a 1 fU f A (a) fU m 2 ma m 1 a exp for a 0
2ma 2m Similarly, for B
f B (b) 1 b
exp 2mb 2m for b 0 These are Chi-square distribution with one degree of freedom.
Now, taking b k a , k a 1
1 a 2m
k fK ( k ) 1 k 2m
1 / 2
1 / 2
e a k a da
k Then taking r a / k i.e. da kdr
1 fK ( k ) 1 k 2m
1 / 2
1 / 2
e r 1 r dr 2m
0 The integral part of this equation is the Beta function B 1 ,1 .
2 Hence f K (k ) B 1 ,1 22 1 1
2 2 1 1 k 2m
e is a Chi-square distribution with two degrees of freedom.
2m 7. Concluding Remarks In this lecture, different statistical properties of one function of two random variables, viz.
sum and difference of independent Normal variates, maximum, minimum product and
quotient) are discussed. It is important to remember that the sum of two independent
Gaussian random variates also possesses a normal density function with mean and variance
derived a sum and square of the sum of individual mean and variance respectively. The same
is true for the difference of two independent Gaussian random variates. Also to note that the
expression for quotient of two random variables can be derived using the same principles
followed in the case of product of two random variables. Two functions of two random
variables will be discussed in next lecture, to explain the theories further....
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This note was uploaded on 03/18/2014 for the course CE 5730 taught by Professor Dr.rajibmaity during the Spring '13 term at Indian Institute of Technology, Kharagpur.
- Spring '13
- Civil Engineering