We’re going to look at a random process, which is a sequence of random process, which is a sequence of random variables that depend on time. Let $X(t)=A^t$ where $A$ has the density $f_A (a)=(3/8) a^2$ for $0\le a\le 2$. Assume $t\ge 1$.
a) Find $F_X (x)$ and $f_X (x)$ (to get this, assume $t$ is fixed-so you can assume the $t$ is a constant).
b) If $t=2$ find $P(X>1)$.
c) Find $E(X)$
Progress
For a) I got $F_X(x)=(xt)^3/8-x^3/8$ from $0≤x≤(2/t)$ and $f_X (x)=(3/8)x^2t^3-(3/8)x^3$ and c) I got $3/2t-3/2t^4$. Are those right and what do I do for b)?
Here is my work for a.
$F_X (x)=P(X≤x)=P(A/t≤x)=P(A≤xt)$
$=∫_x^xt〖3/8) a^3 (da)=1/8 a^3 |_x^xt 〗$
$0≤a≤2→0≤xt≤2$
$=1/8 (xt)^3-1/8 x^3$
$0≤x≤2/t$
$f_x (x)=d/dx (F_x (x))$
$=3/8 x^2 t^3-3/8 x^2$
$0≤x≤2/t$
The notation $F_X$ is absurd since $X$ is a random process, not a random variable. For every $t$ and $x$ positive, $[X(t)\leqslant x]=[A^t\leqslant x]=[A\leqslant x^{1/t}]$ hence $F_{X(t)}(x)=F_A(x^{1/t})=\min\left\{1,\frac18x^{3/t}\right\}$. Equivalently, $F_{X(t)}(x)=0$ for every $x\leqslant0$, $F_{X(t)}(x)=\frac18x^{3/t}$ for every $0\leqslant x\leqslant2^t$ and $F_{X(t)}(x)=1$ for every $x\geqslant2^t$.
You might want to continue.