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Let $V_1$,$V_2$, $r$ be independent random variables, where $V_1$, $V_2$ are Gausian with the same distribution and $r$ is uniformly distributed in $[0,1]$ if

$X(t)=V_2 I(r \geqslant t)+V_1 I(r

a)find the mean of this process:

For the expectation I have the following:

$\mathbb{E}X(t)=μ(1-t)+μ t=μ$

b) find the autocorrelation function of this process

$R(t,s)=\mathbb{E}( X(t) X(s) )$

I have simplified this as much as I can, I don't know why I can't copy and paste what I have done, when I try only some of it appears. I have simplified it down to $\sigma^2$ * probability of union of the indicator functions and a similar expression with the mean.

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    @Sasha: Thanks, nice to know.2012-05-20

1 Answers 1

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Let $\mu$ denote mean of Gaussians $\mu = \mathbb{E}(V_1)=\mathbb{E}(V_2)$. $ \begin{eqnarray} \mathbb{E}\left( X(t) \right) &=& \mathbb{E}\left( V_2 I(R \geqslant t) + V_1 I(R As to the autocorrelation function: $\begin{eqnarray} \mathbb{E}\left( X(t) X(s)\right) &=& \mathbb{E}(\mathbb{V_2^2}) \mathbb{E}\left(I(R \geqslant t)I(R \geqslant s)\right) \\ &+& \mathbb{E}(\mathbb{V_1 V_2}) \left( \mathbb{E}\left(I(R \geqslant t)I(R < s)\right) + \mathbb{E}\left(I(R < t)I(r \geqslant s)\right) \right) \\ &+& \mathbb{E}(V_1^2) \mathbb{E}(I(R>t)I(R>s)) \end{eqnarray} $ Now expectations of product of indicator functions is done by converting them to probabilities and doing some logic: $\mathbb{E}\left(I(R \geqslant t)I(R \geqslant s)\right) = \mathbb{P}(R \geqslant t, R\geqslant s) = \mathbb{P}(R \geqslant \max(t,s))$ $\mathbb{E}\left(I(R\geqslant t) I(R R\geqslant t)$ $\mathbb{E}\left(I(r\geqslant s) I(R R\geqslant s)$ $ \mathbb{E}\left( I(R < t)I( R Hence $\begin{eqnarray} R(t,s) &=& \mathbb{E}(V^2) \left( \mathbb{P}\left(R \geqslant \max(t,s)\right) + \mathbb{P}\left(R < \min(t,s)\right) \right) + \mu^2 \mathbb{P} \left( \max(t,s) > R \geqslant \min(t,s) \right) \\ &=& \mathbb{E}(V^2) + \left(\mu^2 - \mathbb{E}(V^2)\right) \mathbb{P} \left( \max(t,s) > R \geqslant \min(t,s) \right) \end{eqnarray} $

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    i solved the question but the solution my professor gave me has confused me,i have solved it using this method and i am working through the method he used ie evaluate the 2 cases, case 1 0≤t≤s≤1 etc, he has P(r2012-06-05