→ bmka:先自己做做看 10/06 03:15
1. A random variable X has uniform distribution on interval [a, b]. Show
that Var(X) =1 /3(b^2 + ab + a^2) -1/4 (a + b)^2. You may want to use the
fact
Var(X) = E(X^2) - E(X)^2.
2. Suppose that the joint p.d.f. of two random variables X and Y is as
follows:
f(x; y) = 1/8c(x + y) for 0<= x <= 1 and 0 <= y <= 2
0 otherwise
a) Find the value of constant c.
b) Determine the conditional p.d.f of X for every given value of Y.
c) P(X < 1/2│Y = 1/2) and E(X│Y = 1=2).
d) Are X and Y independent? Why or why not?
e) It turned out that E(X) = 5/9, E(X^2) = 7/18, E(Y ) = 11/9, and E(Y^2) =
16/9.
Compute E(XY ). Given these, what is Var(2X - 3Y + 8)?
** Hint: for e) and question 2 c) in the below, use the fact V ar(Σi aiXi +
c) =
Σi a^2 V ar(Xi) +ΣijΣ aiajCov(Xi;Xj). Note that since aiaj = ajai, it can
also be written asΣi a^2 V ar(Xi) + 2ΣΣi>jaiajCov(Xi;Xj), which implies
V ar(aX +bY +c) = a^2V ar(X)+b^2V ar(Y )+2abCov(X; Y ) and V ar(aX +bY +cZ
+d) =
a^2V ar(X) + b^2V ar(Y ) + c^2V ar(Z) + 2abCov(X; Y ) + 2bcCov(Y;Z) +
2acCov(X;Z).
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※ 編輯: briliant 來自: 68.49.45.90 (10/05 21:38)
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