Beta distribution - mean vs mode

#1
I'm new to probability distributions and I'm trying to understand something about the Beta distribution. Suppose I am practicing kicking field goals and I want to understand the probability of successful attempts. If I create a Beta distribution like this:

Beta(20,10)

Where 20 is the number of failed attempts and 10 is the number of successful attempts (for a total of 30 attempts). Given this, I was successful on one-third of the attempts and indeed the mean of this distribution (as I've computed it) is 0.3333. However, the mode (as I've computed it) is 0.32142857142857145.

My question is, why is the mode (the value at which the PDF is maximized) less than the mean? If I represent this experiment with a binomial distribution Binomial(n = 30, p = 1/3), the mean and the mode are equal.

Conceptually, it's easier for me to understand the Binomial distribution as it seems really straight-forward: Out of 30 tries, I made 10 field goals. But with the Beta distribution, I'm not sure I really understand what it is saying, hence I don't understand why the mode and mean are different. I understand that the Binomial distribution is discrete and the Beta distribution is continuous, so I'm assuming that has something to do with it.....
 

Dason

Ambassador to the humans
#2
Why would the mean and mode have to be equal? And even take your binomial example. Try using p = 0.33 instead of exactly 1/3. Now the mean is 9.9 and the mode is still 10. In general it's very rare for the mean and mode to be exactly equal (unless you're looking at a theoretical normal distribution).
 
#3
As I understand it, the mean is the expected value over the long-run and I would have thought that the expected value would be where the likelihood is maximized (in the case of a unimodal distribution, at the mode....)
 

Miner

TS Contributor
#4
Conceptually, the mean is the center of mass (balance point) of the distribution. If the distribution is symmetrical, the mean and mode are the same. However, when the distribution is skewed, the mean is pulled toward the longer tail, while the mode remains nearer the short tail. The greater the skew, the greater the difference.

1566850811070.png
Images are exaggerated to show the differences.

As I understand it, the mean is the expected value over the long-run and I would have thought that the expected value would be where the likelihood is maximized (in the case of a unimodal distribution, at the mode....)
The exponential distribution is an extreme example, but the mean is at the ~67th percentile while the mode is almost zero.