I have been given two random variables $X$ and $Y$ following exponential with means $\lambda_1$ and $\lambda_2$. Let $Z_1 = min(X,Y)$ and $Z_2$ will be 0 if $Z_1 = X_1$ and it will be 1 if $Z_1 = X_2$.

It is required to Find mle of $\lambda_1$ and $\lambda_2$ based on sample of size n on $Z_1$ and $Z_2$.

I can derive the distribution of $Z_1$ as:

$F(z_1) = 1 - e^{-\frac{\lambda_1 + \lambda_2}{\lambda_1 \lambda_2}z_1}$.

Also, for deriving the distribution of $Z_2$, I can see that:

$P(Z_2 = 0) = P(X < Y) = \frac{1}{2} = P(Z_2 = 1)$.

Hence, $Z_2$ can be thought of as Binomial distribution with $p = \frac{1}{2}$.

Now, I am not sure how can I make the Likelihood equation in this case? Specially, How can I make the likelihood equation in two distinct distributions. I could just multiply they if they were independent but in this case we can see that $Z_1$ appears in the distribution of $Z_2$ which means we cannot just multiply the likelihoods of $Z_1$ and $Z_2$ as they are non-independent. Any hints would be appreciated. Thanks in advance.


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