Determine the distribution of $Y_i$

Question

Let $X_1, \ldots , X_n$ be independent, identically distributed random variables with $$P(X_i=-1)=P(X_i=1)=\frac{1}{2}$$ We consider the random variables $Y_i=\max \{X_i,X_{i+1}\}$, $i=1,\ldots , n-1$.

(a) Determine the distribution of $Y_i$, $i=1,\ldots , n-1$.

(b) Calculate the expected value of $Y_i$, $i=1,\ldots , n-1$.

(c) Calculate the covariance of $Y_i$ and $Y_j$, i.e. $\text{Cov}(Y_i, Y_j)=E(Y_iY_j)-E(Y_i)E(Y_j)$, $i,j=1,\ldots , n-1$.

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For question (a) the r.v. $Y_i$ has the same distribution as the r.v.s $X_i$, or not?

Answer 1

For question (a) the distribution of $Y_i$ is not the same as the distribution of $X_i$ because we have four possible outcomes for $(X_i, X_{i+1})$ which all have the probability $\frac{1}{4}$: $$(-1,-1), (1,-1), (-1,1), (1,1)$$ Taking the max of both values we get 1 in three of four cases and -1 in only one case. Therefore $$P(Y_i = 1) = \frac{3}{4}, \ P(Y_i=-1) = \frac{1}{4}$$

Calculating the expected value with this should be no problem.

I hope I could help you.