A Function of Two Random Variables
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1 akultät Inormatik Institut ür Sstemarchitektur Proessur Rechnernete A unction o Two Random Variables Waltenegus Dargie Slides are based on the book: A. Papoulis and S.U. Pillai "Probabilit random variables and stochastic processes" McGraw Hill 4th edition
2 unction o Two Variables Given two random variables and and a unction g we would like to epress a new random variable as g This ma signi i or eample the energ cost o a server that can be epressed as the cost o processing and communication. It is thereore important to epress the PD o in terms o the joint PD.
3 unction o Two Variables Some o the additional relationship we are concerned are displaed d below. + ma min g + / tan 1 / 3
4 unction o Two Variables The distribution o is epressed as ξ P g P[ D ] P D dd D D 4
5 Addition Suppose +. ind the distribution and ddensit o. The region o D in the plane is displaed b the shaded region to the let o the line P dd 5
6 Addition To compute the PD o we appl Leibnit s it dierentiation rule. I b H a h d then dh db da h d d h b b h a + d a Hence the PD o can be computed as + + d d d + + d. d. Alternativel it can also be epressed in terms o the PD o 6
7 Addition I and are independent then Inserting this equation in the previous epression ields: + d + d. The above epression is the standard d convolution o two unctions namel and. 7
8 Addition As a Special case i or < and or > the new limit it o D is set as shown below. dd d d > d. Alternativel in the case o independence: > d d 8
9 Subtraction Suppose -. ind the distribution and ddensit o. The region o D in the plane is displaed b the shaded region to the let o the line P dd 9
10 Subtraction The densit o is given b d d d d. d + I and are independent then + + d As a special case i < and then can be either positive or negative. < This requires the analsis o the problem into two separate conditions or > and <. 1
11 Subtraction dd dd + + d + + d <. 11
12 Multiplication Suppose /. ind the distribution o. Our aim is to ind and epression or P /. The inequalit / can be epressed as i > and i <. Hence need to be conditioned b the event A > and its compliment A. { / } { / A A } { / A} { / A} A > A < 1
13 Multiplication Using the propert o mutuall eclusive: P / P / > + P / < P > + P <. Integrating over the two regions results + dd + dd. Dierentiating with respect to gives + d + d + d < < +. 13
14 Multiplication Note that i and are nonnegative random variables then the area o integration reduces to that shown below. As a result dd dd + d > otherwise. 14
15 Circle Suppose +. ind the distribution and ddensit o. The distribution o is epressed as. P + dd + But the equation + represents the area o a circle with a radius. Thereore + 15
16 Circle Hence. dd Ater appling Leibnit's distribution:. 1 + d 16
17 Suppose and ddensit o. +. Circle ind the distribution In this case the equation represents a circle o radius. Hence. dd Dierentiating the above epression results + d. 17
18 Ma and Min Suppose ma. Determine the distribution ib ti and densit o. The unctions ma and min are nonlinear unctions. > ma > P > P Combined 18
19 Ma and Min As a result: [ ] ma P P P P + > > I the two random variables are. P I the two random variables are independent then In which case the densit can be epressed as. + 19
20 Ma and Min Suppose min. In this case min >. The distribution o is given as P min P[ > ]. > Combined
21 Ma and Min The distribution o is given b 1 1 P P + > > > I the two random variables are I the two random variables are independent then the densit o becomes becomes. + 1
22 Ma and Min [ ]. inall suppose min / ma Deine the distribution ib ti and densit o. Although represents a complicated unction b partitioning the whole space as beore it is possible to simpli this unction. As beore / / >. + > P + P >. P P / + P / >
23 Ma and Min I and are both positive random variables then < <1. 1 The two terms o the above equations are shown b the shaded regions below > dd + dd. { + } d + d + d 3
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![Joint ] X 5) P[ 6) P[ (, ) = y 2. x 1. , y. , ( x, y ) 2, (](/thumbs/89/97775493.jpg "Joint ] X 5) P[ 6) P[ (, ) = y 2. x 1. , y. , ( x, y ) 2, (")
Two-dimensional Random Vectors Joint Cumulative Distrib bution Functio n F, (, ) P[ and ] Properties: ) F, (, ) = ) F, 3) F, F 4), (, ) = F 5) P[ < 6) P[ < (, ) is a non-decreasing unction (, ) = F ( ),,,
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