Is there a way to apply Bayes rules to this question?

Densities and conditioning on an event. Let $X$ be a random variable with PDF $$ f_ X(x)=\begin{cases} x/4,& \text {if } 1<x\leq 3,\\ 0,& \text {otherwise}, \end{cases} $$ and Let $A$ be the event that $\{ X\geq 2\}$.
Find $f_{X|A}(x)$.

I know that $f_{X|A}(x)$ = $f_{X}(x)/P(A)$

We can calculate $P(A) = 5/8$ by integrating or through area. What I want to know is. Is it possible to make it like this $$ f_{X|A}(x)*P(A) = f_{X}(x)*P(A|x) $$

Solution 1:

What I want to know is: is it possible to make it like this $$f_{X|A}(x)\cdot P(A) = f_{X}(x)\cdot P(A|x)$$

Yes. Since: $~\mathsf P(A\mid X{\,=\,}x) = \mathbf 1_{2\leqslant x}~$ then:

$$\begin{align}f_{X\mid A}(x) &= \cfrac{f_X(x)\,\mathbf 1_{2\leqslant x}}{\mathsf P(A)}\\[1ex]&=\dfrac 25\,x\,\mathbf 1_{2\leqslant x\leqslant 3}\\[1ex]&=\begin{cases}2x/5&:& 2\leqslant x\leqslant 3\\0&:& \text{otherwise}\end{cases}\end{align}$$