consistency of maximum likelihood estimator For population with n size and following density function
$$f(y, a)= (1/6a^4)y^3e^{-y/a}$$
For that, I have found the maximum likelihood estimator of a which is $\hat{a}= \bar{y}/4$
I have also shon that this is unbiased estimator of a.
But I cannot show whether this MLE estimator is consistent.
Please help me to do that.
I know that if $lim(|\hat{a}-a|>c)=0$ then MLE is consistent estimator.
I do not know how I can show this limit.

An unbiased estimator is by definition also consistent, so you have actually already shown that.
To see it directly, you just have to show that $a=E[y]/4$, where $E[y] = \int yf(y,a)$ .
(This is because $\lim_{n \to \infty} \bar y = E[y]$, due to the law of large numbers)
