Joint distribution, independent random variables

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E(X) = 1/2*1 + 1/2*5=3
E(Y) = 3/8*(-4) + 3/8*2 + 1/4*7=1
COV(X,Y) = 1/8*(1-3)*(-4-1)+1/4*(1-3)*(2-1)+...

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5:44
A fair coin is tossed four times. Let X denote the number of heads occuring and let Y denote the longest string of heads occuring. (i) determine the joint distribution of X and Y . (ii) find Cov(x,Y)and rho(X,Y) ( dit laatste is de correlatiecoefficient van (X,Y) )
We beginnen met the finite sample space S
S={(HHHH)(THHH)(HTHH)(HHTH)(HHHT)(HHTT)(HTHT)(HTTH)(THHT)((THTH)(TTHH)(TTTH)(TTHT)(THTT)(HTTT)(TTTT)}
aantal elementen van S =2.2.2.2=16 elementen

[aadkr]

X(S)={0,1,2,3,4}
Y(S)={0,1,2,3,4}
Product set: X(S)xY(S)={(0,0)(0,1)(0,2)(0,3)(0,4)(1,0)(1,1)(1,2)(1,3)(1,4)(2,0)(2,1)(2,2)(2,3)(2,4)(3,0)(3,1)(3,2)(3,3)(3,4)(4,0)(4,1)(4,2)(4,3)(4,4)} Dit zijn 25 elementen.
Wordt vervolgd

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De stochasten X en Y zijn afhankelijk.

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5:45
Two cards are selected at random from a box which contains five cards numbered 1,1,2,2,3
Let X denote the sum and Y the maximum of the two numbers drawn. (i) Determine the joint distribution of X and Y.
(ii) Find Cov(X,Y) and rho(X,Y) ( dit laatste is de correlatiecoefficient).

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Zorg dat de 5 kaarten te onderscheiden zijn.
1a,1b,2a,2b,3.
We trekken in feite een combinatie. Aantal combinaties 2 kaarten uit een verzameling van 5 elementen is (5 boven 2)=10
Finite Sample Space is
S={(1a,1b), (1a,2a),(1a,2b),(1a,3),(1b,2a),(1b,2b),(1b,3),(2a,2b),(2a,3),( 2b,3)}
(dit zijn10 elementen.)

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wordt vervolgd