How Does the MGF Indicate Equal Likelihood in Variable Distribution?

[fisher garry]

I don't understand what they are doing here. They start with the mgf for the binomial which I understand. But what is $E[e^{tX}]$? The average of the binomial mgf? And finally why does this explain that X is equally likely to take on any of the values 0,1,..,n?

 

[andrewkirk]

$E[e^{tX}]$ is the Unconditional mgf of $X$. Contrast that with the first line in the proof, which gives the Conditional mgf of $X$.

The above shows that the Unconditional mgf is identical to the mgf of a random variable that is equally likely to take any of the values 1, 2, ..., n.

It is a theorem of probability theory that a mgf is a complete specification of a distribution, so if two random variables have the same mgf, they have the same distribution.

Hence, $X$ has the distribution of a random variable that is equally likely to take on any of the values 1, 2, ..., n. Hence, $X$ is equally likely to take on any of the values 1, 2, ..., n.