Assume $X_1, \ldots, X_n$ are i.i.d exponential($\lambda_x$), independent of $Y_1, ..., Y_m$, which are also i.i.d exponential ($\lambda_y$). Here exponential($\lambda$) corresponds to the density function $f(x) = \lambda e ^ {-\lambda x}$ for $x \geq 0$. From this, I want to find the likelihood ratio statistic for the null hypothesis $H_o: \lambda_x = \lambda_y$ versus the alternative hypothesis, $H_a : \lambda_x \neq \lambda_y$.
My approach so far: Since the LR statistic is given by the ratio of the maximum of the likelihood function under $H_o$ to the likelihood function overall, this is the expression I came up with for it:
\begin{align*} \Lambda = \frac{\max \left( \prod_{i=1}^{n} \lambda_x e^{-\lambda_x x_i} \cdot \prod_{j=1}^{m} \lambda_x e^{-\lambda_x y_j} \right)}{\max \left( \prod_{i=1}^{n} \lambda_x e^{-\lambda_x x_i} \right) \cdot \max \left( \prod_{j=1}^{m} \lambda_y e^{-\lambda_y y_i} \right)} \end{align*}
Is this the right expression for the Likelihood ratio statistic? I would appreciate any help on this?