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Let $X_1, X_2,.... X_{n+1}$ be a sequence of independent identically distributed random variables taking the value $1$ with probability $p$ and $0$ with probability $1-p$. Let $Y_k = 0$ if $X_k + X_{k+1}$ is even and $Y_{k} = 1$ if $X_k + X_{k+1}$ is odd. The aim is to find the expectation and variance of $Y_1 +....+ Y_n$. This is one of the statistics qualifying exam problem from previous years and I am just curious as to how to begin solving it. Any help, however small, is much appreciated

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Hints:

  • For every $k$, $\mathrm E(Y_k)=\mathrm P(Y_k=1)=\mathrm P(X_k\ne X_{k+1})=q$, where $q$ is defined as $q=\mathrm P(X_1\ne X_2)$.
  • For every $k$, $\mathrm E(Y_k)=\mathrm E(Y_k^2)$.
  • For every $k$ and $i$ such that $|k-i|\geqslant2$, $\mathrm{Cov}(Y_k,Y_i)=0$.
  • For every $k$, $\mathrm{Cov}(Y_k,Y_{k+1})=\mathrm P(Y_k=Y_{k+1}=1)-q^2$.
  • For every $k$, $\mathrm P(Y_k=Y_{k+1}=1)=\mathrm P(X_k\ne X_{k+1},X_{k+1}\ne X_{k+2})=r$, where $r$ is defined as $r=\mathrm P(X_1\ne X_2,X_2\ne X_3)$.

Now, you might be able to compute $q$ and $r$ as functions of the parameter $p$ and to collect all these basic facts to reach the expectation and the variance of $Y_1+Y_2+\cdots+Y_n$.

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