Most probable value vs maximum of the distribution Given a distribution p(x), there are two things that can be calculated.

*

*Value of x for which p(x) is maximum.

*Most probable value of x weighted over p(x).

Would these two values of x be the same?

My answer might not be a full-fledge solution but I think it should still help.
Clearly, the most probable value of x weighted over $P(x)$ is an expected value $E[X]$. Now, assume that what you claim is true. Let's say that there is a discrete probability where $P(x=12)=0.2$ and $P(x=2)=0.8$ Clearly, the value of $x$ for which $P(x)$ is maximum is $2$ but the expected value is $E[X]=\sum^2_{i=1}P(X=x_i)x_i=P(x=2)*2+P(x=12)*12=0.8*2+0.2*12=4.0$. Well, $12\ne4$ and I would argue that even though $4.0$ is closer to $2$ than $12$, you can't just say that these values are the same
