mle means that maximizer of joint probability mass function.
I want to know the
mle of theta (4) and
Mle of theta (2)
I dont know what it means (2) and (4) What i ask you is not same with the question of picture i posted but the data is same. Is it means that when x=2 or x=4 ? Then mle of theta(2) is 3?
The question is a bit imprecisely stated, because it is not clear whether the MLE of $\theta$ is based on a single observation $X = x$, or a sample $\boldsymbol X = (x_1, x_2, \ldots, x_n)$.
For the time being, assume the former case; that is to say, we observe a single outcome $X = x$, and we wish to find $\hat \theta$ such that the likelihood $\mathcal L(\theta \mid x)$ is maximized when $\theta = \hat \theta$. More concretely, suppose you observed $X = 1$. Then what value of the parameter $\theta$ maximizes the likelihood of having observed that outcome? Clearly, $\theta = 3$ is impossible, for if $\theta = 3$, $\Pr[X = 1] = 0$. So, if you read the first row of the table, you'd see that the choice $\hat \theta = 1$ in the case that one observes $X = 1$ is the maximum likelihood estimator.
From this, we realize that the MLE $\hat\theta$ will be a piecewise relation with respect to a single outcome of $X$; moreover, it is not necessarily unique. For example, if we observed $X = 4$ (last row of the table), for what $\theta$ is this observation most likely? Either $\hat \theta = 1$ or $\hat \theta = 3$ results in $\Pr[X = 4] = 1/2$, so the MLE in this case is either $1$ or $3$.
If we put all of this together, we can write $$\hat \theta \in \begin{cases} 1, & X = 1 \\ 3, & X = 2 \\ 2, & X = 3 \\ \{1, 3\}, & X = 4. \end{cases}$$ All we did was take each row and locate the column with the largest value in that row. And because the last row had two values that were maximal, either choice would maximize the likelihood.
If, however, we observe a sample comprising multiple independent realizations of $X$ from this distribution, then the calculation of the MLE is considerably more complicated; e.g., if you observed the sample $$\boldsymbol X = (1, 2, 1),$$ you would know that $\theta \in \{0, 2\}$, since $\theta \in \{1, 3\}$ would not allow observing both $X = 1$ and $X = 2$. Then it is easy to see that of the two such choices for $\theta$, $\hat \theta = 2$ is the MLE because $1/27 > 1/64$. But for more general samples and more general sample sizes, you'd find it more difficult to write a general expression for $\hat \theta$ in terms of the sample.