$\{X(t),t>0\}$ be a random variable and have $Y\sim U(0,\pi )$ distribution. Let $X(t)=e^{2Y}t$. Find autocorrelation function of $X(t)$.
$Y\sim U(0,\pi )\Rightarrow f_Y(y)=\frac 1 \pi $ where $0\le y\le \pi$ then
$\begin{align}R_{X}(t_{1},t_{2})&=\Bbb E\left [ X({t_{1}})X(t_{2}) \right ]\\[1ex]&=\Bbb E\left [ e^{2Y}t_{1}\cdot e^{2Y}t_{2} \right ]\\[1ex]&=\Bbb E\left [ e^{4Y}\cdot t_{1}t_{2} \right ]\\[1ex]&=\int_{0}^{\pi }\frac{1}{\pi }e^{4y}(t_{1}t_{2})dy\\[1ex]&=\frac{t_{1}t_{2}}{4\pi }(e^{4\pi }-1)\end{align}$
Is my solution okay? Any answer will be appreciated.