[問題] 機率的問題

[goshagainfuc (...)]

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?

--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 59.117.67.169
推 ym7226:1.不對喔再看看 2.展開看看 3.機率論最前面的東西吧
推 TOOYA:1 你為什麼前面ab小題互換 後面又換回來??
推 TOOYA:2 減一項加一項E(Y|X)之後展開 中間項取E(.)=E(E(.|X))即可
推 TOOYA:3 Bn=(A-An)∪(An-A) Bn->ψ as n→∞ imply what u want
推 goshagainfuc:感謝 我再試看看 囧