Question

This problem is about Probability.

Problem 4. Let X(t),t >0 be a random process defined as X(t) e-Yt,t> 0 where Y~Unif(0,1) (a) Find the PDF fx(t) of the one-dimensional distribution X(t) (b) Find E(X(t)),t> 0 (c) Find Cx(t,t+s), where s,t > 0

Please explain every thing.
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Answer 1

Given $X\left ( t \right )=e^{-Yt};t>0\Rightarrow Y=-\frac{1}{t}\ln X$ and

Y~Unif (0,1)

a) The PDF of $V=g\left ( U \right )$ is $f_V\left ( v \right )=f_U\left ( g^{-1}\left ( v \right ) \right )\left | \frac{\mathrm{d} g^{-1}\left ( v \right ) }{\mathrm{d} v} \right |$ . Thus

In x xt

b) The expectation,

E (X (t)) = E(e-rt) E (X (t),-| eytdy E(x(t) )=-(1-e t) .1 0

c) Autocovariance is given by

$C_X\left ( t_1,t_2 \right )=E\left [ \left ( X\left ( t_1-\mu _X \right ) \right ) \left ( X\left ( t_2-\mu _X \right ) \right )\right ]$

Now,

Cy (t, t + s) = E [e-Y(2t+s+Zux) 1 0 t,t+s) = 2t+ s +