Stats - mle poisson distribution -- quick question

[binbagsss]

This is probably a stupid question , but,

It's easy enough to show that the mle of a poission distribution is $\bar{x}$: $\hat{ \lambda}= \bar{x}$

But,I'm then looking at the generalized ratio test section of my book, multinomial, it esitmates $\lambda$ for some data by $\sum \frac{( number X frequency that number occurred)}{ n}$

how are these 2 equivalent?

Thanks in advance.

 

[haruspex]

This is probably a stupid question , but,

It's easy enough to show that the mle of a poission distribution is $\bar{x}$: $\hat{ \lambda}= \bar{x}$

But,I'm then looking at the generalized ratio test section of my book, multinomial, it esitmates $\lambda$ for some data by $\sum \frac{( number X frequency that number occurred)}{ n}$

how are these 2 equivalent?

Thanks in advance.

It's the same thing. Suppose you have a sample 9, 5, 7, 8, 9, 9. You can get the sum by just adding them, or you can rewrite it as 1 x 5 + 1x 7 + 1 x 8 + 3 x 9.