Process' mean, covariance and stationarity

Question

Let $X_t=Yt+Zt^2$ be random process, where $Y$,$Z$ are uncorrelated random variables, with characteristics: $EY=3$, $EZ=0.5$, $DY=1$, $DZ=0.05$. Find $X_t$ mean and covariance and prove whether process is wide-sense stationary.

Covariance:

$\Gamma(Y,Z)=E(YZ)-EYEZ=EYEZ-EYEZ=0$

How about its mean? Any hint?

Hint as requested: Linearity of expectation! $E[aX+bY] = aE[X]+bE[Y]$ where $a$ and $b$ are real numbers. — Dilip Sarwate, May 27 '13 at 19:11
Well, not quite. You have the idea down correctly, but $EZ=0.5$ in your problem statement while you used $EZ=0.05$ in the answer you worked out. One of these is a typographical error. — Dilip Sarwate, May 27 '13 at 19:23
Thanks. Can I assume, that this process isn't wide-sense stationary -- ($EX_{t_1} \neq EX_{t_2}$)? — atomoutside, May 27 '13 at 20:06
Yes, the fact that $EX_{t_1}\neq EX_{t_2}$ is sufficient to prove that the process is not stationary in the wide sense. — Dilip Sarwate, May 27 '13 at 21:06

score 1 · Answer 1 · answered May 27 '13 at 20:12

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$\newcommand{\cov}{\operatorname{cov}}$

You've seen the answer to the first question in comments. As for covariance, you're making it unreasonably complicated. $$ \begin{align} \cov(X_t,X_s) & = \cov(Yt+Zt^2,Ys+Zs^2) = \cov(Yt,Ys+Zs^2)+\cov(Zt^2,Ys+Zs^2) \\[8pt] & =\cov(Yt,Ys)+\cov(Yt,Zs^2)+\cov(Zt^2,Ys)+\cov(Zt^2,Zs^2) \\[8pt] & = st\cov(Y,Y) + ts^2\cov(Y,Z) + t^2 s\cov(Y,Z) + t^2 s^2 \cov (Z,Z) \\[8pt] & = st\operatorname{var}(Y) + 0 + 0 + t^2s^2 \operatorname{var}(Z). \end{align} $$

answered May 27 '13 at 20:12

Michael Hardy

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So the fact, that $Y$ and $Z$ are uncorrelated, is useless? – atomoutside May 27 '13 at 20:24
@atomoutside : Why would you say that? I used that fact above. That's where the two $0$s came from. – Michael Hardy May 27 '13 at 20:27
Just noticed uncorrelation -- I first typed and later thought about $0$s. Am I on right path for proving non-wide-sense stationarity: ($EX_{t_1} \neq EX_{t_2}$) ? – atomoutside May 27 '13 at 20:33

Process' mean, covariance and stationarity

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