How to Find an expression to the probability that the number of Passing cars in all the routes combined is less than $n$ during $t$ seconds

Question

A Road Has $4$ Routes A,B,C,D . The Number of Passing cars in each route is Poision Random variable with parameters $\lambda_A ~~\lambda_B~~\lambda_C~~\lambda_D ~~\frac{Cars}{Second}$.

Find an expression to the probability that the number of Passing cars in all the routes combined is less than $n$ during $t$ seconds .

Answer Attempt :

Let $Y$ be random variable of the number of Passing cars in all the routes combined during $t$ seconds. We see that $Y$~$Poison(\bar{\lambda} t)$

Let $\bar{\lambda}$ The avarge number of cars passing through the $4$ routes combined per second

$\bar{\lambda} = \lambda_A + \lambda_B + \lambda_C + \lambda_D$

$P(Y=y) = \frac{e^{-(\bar{\lambda}t)}(\bar{\lambda}t)^y}{y!}$

So :

$P(Y<n) = \sum_{y=0}^{y=n} ~\frac{e^{-(\bar{\lambda}t)}(\bar{\lambda}t)^y}{y!} = \sum_{y=0}^{y=n} ~\frac{e^{-(\lambda_A + \lambda_B + \lambda_C + \lambda_D)t}~((\lambda_A + \lambda_B + \lambda_C + \lambda_D)t)^y}{y!}$

First is $\bar{\lambda} = \lambda_A + \lambda_B + \lambda_C + \lambda_D~$ or $~\bar{\lambda} = \frac{\lambda_A + \lambda_B + \lambda_C + \lambda_D}{4}$ since $\lambda_{poison}$ is the average

Second is this correct\wrong expression ?

should $\bar{\lambda} = E(Y) = E(x_A) + E(x_B)+ E(x_C) + E(x_D)$

Answer 1

Your answer looks correct to me, apart from a typo which I've taken the liberty of correcting.

And since $\ Y=x_A + x_B+x_C+ x_D\ $, then \begin{align} \overline{\lambda}&=E(Y)\\ &=E\left(x_A\right) + E\left(x_B\right)+E\left(x_C\right)+ E\left(x_D\right)\\ &=\lambda_A+\lambda_B+\lambda_C+\lambda_D\ . \end{align}