The problem is:
In a mile race between A and B, the time it takes A to complete the mile is an exponential random variable with rate $\lambda_a$ and is independent of the time it takes B to complete the mile, which is an exponential random variable with rate $\lambda_b$. The one who finishes earliest is declared the winner and receives $Re^{-\alpha t}$ if the winning time is $t$, where $R$ and $\alpha$ are constants. If the loser receives zero, find the expected amount that runner A wins.
Here is how I solve this: $$E(W_A) = E(W_A|\text{A wins})P\{\text{A wins}\}=\frac{\lambda_a}{\lambda_a+\lambda_b}R\int e^{-\alpha t}\lambda_ae^{-\lambda_at}dt$$
and this is wrong. It is all about how to handle $E(W_A|\text{A wins})$ and in my fault way, I think that since A wins then it is reasonable to set that $t = t_a(\text{the time A finished the race})$, so the distribution of t is equal to that of $t_a$, which is an exponential distribution with rate $\lambda_a$. I don't know how this thinking went wrong.
Instead, the answer is: $$E(W_A) = \frac{\lambda_a}{\lambda_a+\lambda_b}R\int e^{-\alpha t}re^{-rt} dt$$ where $r = \lambda_a+\lambda_b$ and I know that $t=min(t_a,t_b)$ is actually a exponential random variable with rate $\lambda_a+\lambda_b$. I am not sure if the answer is based on a conditional perspective, if it is, for $E(W_A|\text{A wins})$, it should be: $$E(W_A|\text{A wins}) = \int e^{-\alpha t} f_{t|\text{A wins}}(t) dt$$ thus, since there is a condition "A wins", how come $f_{t|\text{A wins}}(t) = re^{-rt}$?
What we seek is $E[\mathbf{1}_{\{M=A\}}Re^{-\alpha M}]$ where $M=\min(A,B)$. This is found as following: $$\begin{aligned}E[\mathbf{1}_{\{M=A\}}Re^{-\alpha M}]&=E[\mathbf{1}_{\{A\leq B\}}Re^{-\alpha A}]=\\ &=R\int_{[0,\infty)}e^{-\alpha a}\int_{[a,\infty)}\lambda_b e^{-b\lambda _b}\lambda_ae^{-\lambda_a a}dbda=\\ &=R\lambda_a\int_{[0,\infty)}e^{-\alpha a}e^{-(\lambda_a+\lambda_b) a}dbda=\\ &=R\frac{\lambda_a}{\lambda_a+\lambda_b}\int_{[0,\infty)}e^{-\alpha a}(\lambda_a+\lambda_b)e^{-(\lambda_a+\lambda_b) a}da=\end{aligned}$$