Let $z _ { n }$ and $w _ { n } ( n = 0,1,2 , \ldots )$ be complex numbers. Consider a bag that contains two red cards and one white card. First, take one card from the bag and return it to the bag. $z _ { k + 1 } ( k = 0,1,2 , \ldots )$ is generated in the following manner based on the color of the card taken.
$$z _ { k + 1 } = \begin{cases} i z _ { k } & \text { if a red card was taken, } \\ - i z _ { k } & \text { if a white card was taken. } \end{cases}$$
Next, take one card from the bag again and return it to the bag. $w _ { k + 1 }$ is generated in the following manner based on the color of the card taken.
$$w _ { k + 1 } = \begin{cases} - i w _ { k } & \text { if a red card was taken, } \\ i w _ { k } & \text { if a white card was taken. } \end{cases}$$
Here, each card is independently taken with equal probability. The initial state is $z _ { 0 } = 1$ and $w _ { 0 } = 1$. Thus, $z _ { n } , w _ { n }$ are the values after repeating the series of the above two operations $n$ times starting from the state of $z _ { 0 } = 1$ and $w _ { 0 } = 1$. Here, $i$ is the imaginary unit.
Answer the following questions.
(1) Show that $\operatorname { Re } \left( z _ { n } \right) = 0$ if $n$ is odd, and that $\operatorname { Im } \left( z _ { n } \right) = 0$ if $n$ is even. Here, $\operatorname { Re } ( z )$ and $\operatorname { Im } ( z )$ represent the real part and the imaginary part of $z$ respectively.
(2) Let $P _ { n }$ be the probability of $z _ { n } = 1$, and $Q _ { n }$ be the probability of $z _ { n } = i$. Find recurrence equations for $P _ { n }$ and $Q _ { n }$.
(3) Find the probabilities of $z _ { n } = 1 , z _ { n } = i , z _ { n } = - 1$, and $z _ { n } = - i$ respectively.
(4) Show that the expected value of $z _ { n }$ is $( i / 3 ) ^ { n }$.
(5) Find the probability of $z _ { n } = w _ { n }$.
(6) Find the expected value of $z _ { n } + w _ { n }$.
(7) Find the expected value of $z _ { n } w _ { n }$.