Germano, i need your help in one last PROBABILITY question. Thank you.?

Well,

okay, as the question is open "for the house" ... let's proceed

X = Sup(X1, X2, X3)

X1 has a uniform distribution within E = {1, 2, 3, 4, 5, 6}

the mutual distribution of (X1, X2, X3) is, for any triplet of

P( X1 = m, X2 = n, X3 = p) = 1/6 * 1/5 * 1/4 = 1/120

by the way, for any two among X1, X2 and X3, for example X1 and X2 we have :

P( X1 = m, X2 = n) = 1/6 * 1/5

P(X = k) means ALL X1 and X2 and X3 are < k

therefore :

of course, we have :

P(X= 1) = 0 and

P(X= 2) = 0 as we draw three numbers without replacement

P(X = 3) = P(X1 = 3, X2 = 1, X3 = 2) + P(X1 = 3, X2 = 2, X3 = 1) + ...

... + two terms for (X2 = 1) + ...

... + two terms for (X3 = 1) + ...

= 6/120

= 1/20

P(X = 4) = P(X1 = 4, X2 = 1, X3 = 2) + P(X1 = 4, X2 = 2, X3 = 1) + ...

... P(X1 = 4, X2 = 1, X3 = 3) + P(X1 = 4, X2 = 3, X3 = 1) + ...

... P(X1 = 4, X2 = 2, X3 = 3) + P(X1 = 4, X2 = 3, X3 = 2) + ...

... + same 6 terms with (X2 =4) + ...

... + same 6 terms with (X3 =4) +

= 3 * 3A2 * 1/120 <--- where 3A2 = 3! / (3 - 2)! = 6

= 18/120

= 3/20

now,

the number of terms (all equal) consists

in choosing 1 among X1, X2 or X3 to represent the MAX and

in arranging the remaining 2 among (4 - 1)

so :

P(X=5) = 3 * 4A2 * 1/120 <---- where 4A2 = 4! / (4 - 2)! = 12

= 36/120

= 6/20

and finally :

P(X=6) = 3* 5A2 * 1/120

= 60/120

= 10/20

summary : we obtained the law :

P(X= 1) = 0

P(X= 2) = 0

P(X=3) = 1/20

P(X=4) = 3/20

P(X=5) = 6/20

P(X=6) = 10/20

and we verify of course that the sum equals 1 !!

the rest is just a matter of applying the formulae...

E(X) = 3*P(X=3) + 4*P(X=4) + 5*P(X=5) + 6*P(X=6)

= (1/20) * (3*1 + 4*3 + 5*6 + 6*10)

= (1/20) * 105

= 21/4 <------------ verify my calc !! I only guarantee the method !!

Var(X) = E(X^2) - E(X)^2

where :

E(X^2) = 3^2*P(X=3) + 4^2*P(X=4) + 5^2*P(X=5) + 6^2*P(X=6)

= (1/20) * (3^2*1 + 4^2*3 + 5^2*6 + 6^2*10)

= 558/20

and you can easily finish !!

et voilà !! ;-)

nice exercise ...

hope it' ll help !!