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Unformatted text preview: . FZ z PZ z f x , y dxdy
X ,Y xy z X and Y to obtain the cdf of In fig. 1, the region for integration of the joint pdf of
Z X Y is given.
4
3
2 Y=Z/X Y 1
0 XY<Z
Y=Z/X 1
2
3
4
20 15 10 5 0
X 5 10 15 20 Fig. 1. The region to be integrated to get the cdf of
product of two random variables X Y So we get, FZ z PZ z f x , y dxdy
X ,Y xy z 0 z / x f X ,Y x , y dy dx f X ,Y x , y dy dx z / x
0 and if t x y , then
FZ z z / x t dt t dt f X ,Y x , dx f X ,Y x , dx zx x x 0 x x /
0 1 t x f X ,Y x , x dxdt z Differentiating this equation with respect to , we get the expression for pdf of the product of
two random variables,
f Z z z 1
1 z
f X ,Y x , dx f X ,Y , y dy x x y y If X and Y are independent random variables, then the product of their marginal can be
substituted for the bivariate pdf, thus we get,
f Z z 1
1 z
z
f X ,Y fY dx f X fY y dy x y
y x 5.2. Quotient of two random variables
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. The expression for CDF is obtained
as,
FZ z PZ z PX / Y z , y 0 P X / Y z , y 0 PX Yz , y 0 PX Yz , y 0 f x , y dxdy X ,Y xy z yz f X ,Y x , y dy dx f X ,Y x , y dy dx yz
0 0 The pdf of the quotient W X / Y of two random variables X and Y is given as, FW w 0 y f X ,Y wy , y dy y f wy , y dy
X ,Y 0 y f wy , y dy
X ,Y If X and Y are independent random variables, then we obtain, FW w y f wy f y dy
X Y Fig. 2. Plot of product and quotient of two random variables
The pdf of the product Z and quotient W of two uniform (0,1) variates is shown in fig. 2.
The dotted line shown in the figure present the common pdf for both X and Y .
On generalization, the product and quotient of statistically independent lognormal varia...
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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
 Dr.RajibMaity
 Civil Engineering

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