7. Two Random Variables

Transcription

1 7. Two Random Variables In man eeriments the observations are eressible not as a single quantit but as a amil o quantities. or eamle to record the height and weight o each erson in a communit or the number o eole and the total income in a amil we need two numbers. Let and denote two random variables r.v based on a robabilit model Ω. Then and d. d

2 What about the robabilit that the air o r.vs belongs to an arbitrar region D? In other words how does one estimate or eamle Towards this we deine the oint robabilit distribution unction o and to be [ ]? [ ] 7- where and are arbitrar real numbers. roerties i + + since we get. 7-

3 3 Similarl we get ii To rove 7-3 we note that or and the mutuall eclusive roert o the events on the right side gives which roves 7-3. Similarl 7-4 ollows.. Ω + +. Ω > +

4 4 iii This is the robabilit that belongs to the rectangle in ig. 7.. To rove 7-5 we can make use o the ollowing identit involving mutuall eclusive events on the right side R ig. 7. R

5 This gives + and the desired result in 7-5 ollows b making use o 7-3 with and resectivel. Joint robabilit densit unction Joint.d. B deinition the oint.d. o and is given b. and hence we obtain the useul ormula 7-6 u v dudv. 7-7 Using 7- we also get + + dd

6 To ind the robabilit that belongs to an arbitrar region D we can make use o 7-5 and 7-7. rom 7-5 and u v dudv + +. Thus the robabilit that belongs to a dierential rectangle equals and reeating this rocedure over the union o no overlaing dierential rectangles in D we get the useul result 7-9 D ig. 7. 6

7 D dd. 7- D iv Marginal Statistics In the contet o several r.vs the statistics o each individual ones are called marginal statistics. Thus is the marginal robabilit distribution unction o and is the marginal.d. o. It is interesting to note that all marginals can be obtained rom the oint.d.. In act Also d d. To rove 7- we can make use o the identit

8 so that +. To rove 7- we can make use o 7-7 and 7- which gives + u dud and taking derivative with resect to in 7-3 we get At this oint it is useul to know the ormula or dierentiation under integrals. Let Then its derivative with resect to is given b Obvious use o 7-6 in 7-3 gives d. H b a h d. b + d d d a dh db da h h b ha d

9 I and are discrete r.vs then i i reresents their oint.d. and their resective marginal.d.s are given b and i i Assuming that i is written out in the orm o a rectangular arra to obtain i rom 7-7 one need to add u all entries in the i-th row. It used to be a ractice or insurance comanies routinel to scribble out these sum values in the let and to margins thus suggesting the name marginal densities! ig 7.3. i i i i i i i m i m i i m i ig n n in mn 9

10 rom 7- and 7- the oint.d. and/or the oint.d. reresent comlete inormation about the r.vs and their marginal.d.s can be evaluated rom the oint.d.. However given marginals most oten it will not be ossible to comute the oint.d.. Consider the ollowing eamle: Eamle 7.: Given constant 7-9 otherwise. Obtain the marginal.d.s and. ig. 7.4 Solution: It is given that the oint.d. is a constant in the shaded region in ig We can use 7-8 to determine that constant c. rom dd c d d cd c c. 7-

11 Thus c. Moreover rom and similarl d d 7- + d d Clearl in this case given and as in it will not be ossible to obtain the original oint.d. in 7-9. Eamle 7.: and are said to be ointl normal Gaussian distributed i their oint.d. has the ollowing orm:. 7- πσ σ ρ e µ ρ σ ρ µ µ µ + σ σ σ + + ρ. 7-3

12 B direct integration using 7-4 and comleting the square in 7-3 it can be shown that + and similarl + d d πσ πσ µ ollowing the above notation we will denote 7-3 as N µ µ σ σ ρ. Once again knowing the marginals in 7-4 and 7-5 alone doesn t tell us everthing about the oint.d. in 7-3. e e µ / σ / σ ~ ~ N µ σ N µ σ As we show below the onl situation where the marginal.d.s can be used to recover the oint.d. is when the random variables are statisticall indeendent.

13 Indeendence o r.vs Deinition: The random variables and are said to be statisticall indeendent i the events { A} and are indeendent events or an two Borel sets A and B in and aes resectivel. Aling the above deinition to the events { } and { } we conclude that i the r.vs and are indeendent then { B} i.e or equivalentl i and are indeendent then we must have

14 I and are discrete-te r.vs then their indeendence imlies or all i. i i Equations give us the rocedure to test or indeendence. Given obtain the marginal.d.s and and eamine whether 7-8 or 7-9 is valid. I so the r.vs are indeendent otherwise the are deendent. Returning back to Eamle 7. rom we observe b direct veriication that. Hence and are deendent r.vs in that case. It is eas to see that such is the case in the case o Eamle 7. also unless In other words two ointl Gaussian r.vs as in 7-3 are indeendent i and onl i the ith arameter ρ. 7-9 ρ. 4

15 Eamle 7.3: Given Determine whether and are indeendent. Solution: Similarl In this case + e otherwise. d e + d and hence and are indeendent random variables. e e e d d