[問題] 機率的問題
1.let X be random variable with probability mass function
n r n-r
P(X=r) = ( )p (1-p) , if r = 0,1,2,..,n. 0≦p≦1.
r
Find the pmfs of the random variables (a) Y=X^2 , and (b) Y=√X
2.show that E(Y-Φ(X))^2 is minimized by choosing Φ(X)=E(Y|X).
3.Let {An} be a sequence of events such that An → A as n→∞.
Show that P(An)→P(A) as n→∞
1.我是寫
X=Y^2 => Y = √X
n √r n-√r
P (r) = P (√r) = ( )p (1-p) , for r = 0 , 1 , 4 , ... , n^2
Y X √r
X=√Y => Y = X^2
n r^2 n-r^2
P (r) = P (r^2) = ( )p (1-p) , for r = 0 , 1 , 2 , √3 , ... , √n
Y X r^2
有那麼簡單嗎?總覺得怪怪的@@""
2.好像是迴歸線的性質?直接算 E(Y-E(Y|X))^2 還有些頭緒 但從E(Y-Φ(X))^2要變出
E(Y|X)真的弄不出來耶@@""
3.我對於這種看似理所當然的命題都不會證耶@@""是因為沒修過高微的關係嗎?
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