If $Y_i$ has a $\textrm{Poisson}(\lambda_i)$ distribution then it has density function, $$p(Y_i|\lambda_i) = \frac{\lambda_i^{Y_i}}{Y_i!}e^{−\lambda_i}$$ Suppose we think the variables $Y_1, \dots , Y_n$ follow a Poisson distribution. For each variable, we have an predictor $X_i$ and would like to model $Y$ as a function of X. So we get the poisson regression model with: $$λ_i = e^{\beta_0+\beta_1X_i}$$ write the log likelihood function in terms of $x,y,\beta_1$ and $\beta_0$.
I am a little stuck with this and think that the likelihood function here would be equal to $$-\sum e^{\beta_0+\beta_1 x_i}-\sum \log(y_i!)+\sum y_i (\beta_0+\beta_1 x_i)$$ Is this correct and if not how would I go about finding this log likelihood function?