$P((X_k,Y_k)=(i,j))=\pi_{ij}$ where $\sum \pi_{ij}=1$ and the frequency of $(i,j)$ is $n_{ij}$.
I'm now asked to find the MLE of $\pi_{ij}$ if $X$ and $Y$ are independent and again if they are not.
I found that the likelihood funciton is: $L(\pi;x)= \prod_{ij} \pi_{ij}^{n_{ij}}$
If $X$ and $Y$ are independent then we can say $\pi_{ij}=\alpha_i \beta_j$ with constraints $\sum \alpha_i = 1$ and $\sum \beta_j =1$ and $\sum \sum \pi_{ij}=1$
Using Lagrange multipliers I find the MLE of $\alpha_i$ and $\beta_j$ are $\frac {2}{\sum_j n_{ij}}$ and $\frac {2}{\sum_i n_{ij}}$
Then the MLE of $\pi_{ij}$ is $\frac {4}{\sum_j n_{ij} \sum_i n_{ij}}$ which doesn't make logical sense to me... Is there some explanation as to why this answer makes sense?
Many thanks for your help!
No, it doesn't make sense. That could even be greater than 1.
Let $N=\sum\sum n_{i,j}$. If they are independent, the mle of $\alpha_i$ is $\frac{\sum_{j}n_{i,j}}{N}$ and the mle of $\beta_j$ is $\frac{\sum_{i}n_{i,j}}{N}$.
Just re-write the likelihood in terms of the $\alpha_i$ and $\beta_j$ to derive those.
$$\prod_{i,j}(\alpha_i \beta_j)^{n_{i,j}}$$ Try to maximize this wrt $\alpha_1$ for example.
It's easier to see if you maximize the log-likelihood.