조건 파악
X_0 - 1 \sim bernoulli(p)
\\[20pt]
\begin{align*}
P(X_0-1 = 0) &= P(X_0=1) = 1-p
\\
P(X_0-1 = 1) &= P(X_0=2) = p
\end{align*}
E(X_0), Var(X_0) 구하기
\begin{align*}
E(X_0) &= 1 \cdot P(X_0=1) +
2 \cdot P(X_0=2) \\
&= 1-p + 2p \\
&= p+1
\\[10pt]
E\big({X_0}^2\big) &=
1 \cdot P(X_0=1) + 4 \cdot P(X_0=2) \\
&= 1-p + 4p \\
&= 3p+1
\\[10pt]
Var(X_0) &= (3p+1) - (p+1)^2 \\
&= 3p + 1 - p^2 - 2p -1 \\
&= -p^2 + p\\
&= -p(p-1)
\\[10pt]
\end{align*}
E(X_1) 구하기
E(X_1) = E\big[E(X_1|X_0)\big]
\\[15pt]
\begin{align*}
E(X_1|X_0 = x)
&= \sum_y y P(X_1=y|X_0=x)
\\[20pt]
E(X_1|X_0 = 1)
&= 0 \cdot P(X_1 = 0 | X_0 = 1) +
0 \cdot P(X_1 = 2 | X_0 = 1) \\
&= 2p~I(X_0 = 1)
\\[20pt]
E(X_1|X_0 = 2)
&= 0 \cdot P(X_1 = 1 | X_0 = 2) +
0 \cdot P(X_1 = 3 | X_0 = 2) \\
&= (1 + 2p)~I(X_0= 2)
\\[25pt]
E(X_1) &= E\big[E(X_1|X_0)\big]
\\&= E\big[2p~I(X_0 = 1) +
(1 + 2p)~I(X_0= 2) \big]
\\
&= 2 E\big[p~I(X_0 = 1)\big] +
(1 + 2p)~E\big[I(X_0= 2) \big]
\\
&= 2p~E\big[I(X_0 = 1)\big] +
(1 + 2p)~E\big[I(X_0= 2) \big]
\\
& = 2p \cdot P(X_0=1) +
(1 + 2p) \cdot P(X_0=2)|
\\
&= 2p\cdot(1-p) + (1+2p) \cdot p
\\
&= 2p - 2p^2 + p + 2p^2
\\
&= 3p
\end{align*}
Var(X_1) 구하기
Var(X_1) = E({X_1}^2) - {E(X_1)}^2
\\[8pt]
E({X_1}^2)
= E\big[E({X_1}^2|X_0=x) \big]
\\[20pt]
\begin{align*}
E({X_1}^2|X_0=x) &= \sum_y~y^2~ P(X_1=y|X_0=x)
\\[20pt]
E({X_1}^2|X_0=1)&=
0 \cdot P({X_1}^2=0|X_0=1) +
4 \cdot P({X_1}^2=2|X_0=1)
\\
&= 4p
\\[15pt]
E({X_1}^2|X_0=2) &=
0 \cdot P({X_1}^2=1|X_0=2) +
4 \cdot P({X_1}^2=3|X_0=2) \\
&= 1\cdot(1-p) + 9p \\
&= 1 + 8p
\\[25pt]
E({X_1}^2) &= E\big[E({X_1}^2|X_0=x)\big]
\\
&= E\big[4p~I(X_0 = 1) +
(1 + 8p)~I(X_0= 2) \big]
\\
&= 4 E\big[p~I(X_0 = 1)\big] +
(1 + 8p)~E\big[I(X_0= 2) \big]
\\
&= 4p~E\big[I(X_0 = 1)\big] +
(1 + 8p)~E\big[I(X_0= 2) \big]
\\
& = 4p \cdot P(X_0=1) +
(1 + 8p) \cdot P(X_0=2)|
\\
&= 4p\cdot(1-p) + (1+8p) \cdot p
\\
&= 4p - 4p^2 + p + 8p^2
\\
&= 4p^2 + 5p
\\[25pt]
Var(X_1) &= E({X_1}^2) - {E(X_1)}^2
\\
&= 4p^2 + 5p - (3p)^2
\\
&= -5p^2 + 5p
\\
&= -5p(p-1)
\end{align*}
Cov(X_0, X_1) 구하기
Cov(X_0,~X_1) = E(X_0X_1) - E(X_0)E(X_1)
\\[15pt]
\begin{align*}
&E(X_0X_1) =
E\big[X_0E(X_1|X_0) \big]
\\[5pt]
&= E\big[X_0\{2p~I(X_0=1) +
(1+2p)~I(X_0=2)\}\big]
\\[5pt]
&= 2p~E[X_0~I(X_0=1)] +
(1+2p)~E\big[X_0~I(X_0=2)\big]
\\[5pt]
&= 2p~E[X_0~I(X_0=1)] +
E\big[X_0~I(X_0=2)\big] +
2p~E\big[X_0~I(X_0=2)\big]
\\[5pt]
&= 2p(1-p) + 2p + 2p\cdot2p
\\[5pt]
&= 2p^2 + 4p
\end{align*}
\begin{align*}
Cov(X_0,~X_1)
&= 2p^2 + 4p - 3p(p+1)
\\[5pt]
&= 2p^2 + 4p - 3p^2 - 3p
\\[5pt]
&= -p^2 + p
\end{align*}
\rho 구하기
\begin{align*}
\rho &= \dfrac{Cov(X_0,~X_1)}{\sqrt{Var(X_0)Var(X_1)}}
\\[20pt]
&= \dfrac{-p(p-1)}
{\sqrt{-p(p-1) \cdot -5p(p-1)}}
\\[20pt]
&= \dfrac{-p(p-1)}
{\sqrt{p(p-1) \cdot 5p(p-1)}}
\\[20pt]
&= \dfrac{1}{\sqrt{5}}
\end{align*}