I'm trying to solve followed first-price auction problem.
Bidder's pdf is $$ f(v_i)= \begin{cases} \dfrac{1}{8}v_i, & \text{if} & 0\leq v_i\leq4\\ 0, & \text{if} & \text{otherwise}\\ \end{cases} $$
Bidders only know their own values. One gets 0 if lose the auction and ($v_i-b_i$) if win the auction game.
The question asks the symmetric bidders' bid under Bayesian Nash equilibrium and expected payoff of bidders and seller.
So, one bidder will maximize as follows:
$$\max_{b_1}\,\,(v_1-b_1)\Pr(b_1>b_2)$$
If we take the derivative with respect to $b_1$ we get
$$b(v_1)=\dfrac{\int v_1f(v_1)\,dv_1}{F(v_1)}$$
We know from question $f(v_1)=\dfrac{1}{8}v_i$ and we can find $F(v_1)=\dfrac{1}{16}v_i^2$. If we replace them, we get
$$b(v_1)=\dfrac{\int v_1\dfrac{1}{8}v_1\,dv_1}{\dfrac{1}{16}v_1^2} = \dfrac{2}{3}v_1$$
So this is the bidding of bidder 1. However, I couldn't find the expected payoff bidder 1 and bidder 2. My question is how can I use PDF or CDF to find the expected payoffs of the bidders.