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If $ X(w,t)=\xi +\eta t $, $\xi$ and $\eta$ are random variable which each of them has normal distribution with parameter $(a,\sigma^{2})$, then compute $P_{t}(X)$ and $K_{X}(t_{1},t_{2})$.

Edit: $\eta$ and $\xi$ are independent parameters. $P$ is distribution function(or CDF) in one dimension. $K$ is covariance.

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Usually when people omit to mention that $\xi$ and $\eta$ are independent, it means $\xi$ and $\eta$ are independent. But it wouldn't hurt to be explicit about that. I'm guess that by $P$ you mean the CDF and maybe by $K$ you mean covariance, but I don't know what the subscript $x$ means. –  Michael Hardy Mar 11 '12 at 20:30
Please clarify what are $P$ and $K$: probability density function and convariance function? If it is homework, please put a tag. –  Ilya Mar 11 '12 at 20:30
Sorry. I have just rewrited my question again. –  demir Mar 11 '12 at 20:36
Forget random processes for a minute. Do you know the definition of covariance of two random variables $W$ and $Z$? Can you find $\text{cov}(W,Z)$ if you know that $W \sim N(\mu_1,\sigma_1^2$, $Z \sim N(\mu_2, \sigma_2^2)$ and their correlation coefficient is $\rho_{W.Z}$? If so, can you figure out the means and variances of $W = \xi + 2\eta$ and $Z = \xi + 3\eta$ and $\text{cov}(W,Z)$? –  Dilip Sarwate Mar 11 '12 at 21:12

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