Re: [理工] [機率]-probability density function
※ 引述《swatch0811 (............)》之銘言：
因為小第我沒修過機率，自己看書後，
又覺得寫出來的答案很不確定，煩請會
的人，幫我解答一下(希望可以大R回復)
謝謝。
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Q1: Let random variables Y = X^2 .Find the probability density
function and expectation of Y for the following cases.
1. X takes the value -2, -1, 0, 1, 2, 3 with
equal probability 1/6.
X -2 -1 0 1 2 3
Y 4 1 0 1 4 9
P(X) 1/6 1/6 1/6 1/6 1/6 1/6
=> Y 0 1 4 9
P(Y) 1/6 1/3 1/3 1/6
P(Y=y) = 1/6 , if y=0
1/3 , if y=1
1/3 , if y=4
1/6 , if y=9
0 , otherwise
2. X is uniformly distributed in the interval (0,1).
f(x)=1 , 0<x<1
y=x^2 x=y^(1/2)
d
f(y) = | ─ y^(1/2) |* f(y^(1/2))
dy x
= 0.5* y(-0.5) *1
Q2: There are two random variables X 屬於 {0, 1, 2} and
┌ 1 , if X = 0
Y = │
└ 0 , if X = 1,2
If P(X=0) = P(X=1) = P(X=2) = 1/3, and P(Y=0) = P(Y=1) =
*有誤 P(Y=2) = 1/2 , then are X and Y orthogonal? uncorrelated?
independent?
orthogonal iff E[XY] = 0
E[XY] = 0 (∵xy=0 for all x,y)
uncorrelated iff cov[XY]=0
cov[XY]= E[XY]-μ_x*μ_y = 0 - 1*1/2 ≠ 0
independent iff P(XY)=P(X)*P(Y) *這部份我不確定orz
P(X=1,Y=1)=0 ≠P(X=1)*P(Y=1)
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