Right now I am totally confused as to the difference between these two distributions. I think theoretical means that a given distribution that we already know its all information. However, for the empirical distribution, we also know all information about it. What is the exactly difference between them?

For in example,

In R,

`dnorm()`

: Obtain the density values for thetheoreticalnormal distribution; why it isn’t an empirical normal distribution?In R,

`density()`

: fit anempiricaldensity curve to a set of values; why in this case, it uses “empirical”?

**Answer**

In an nutshell, when you **know** what the distribution is *and its parameters*, you can construct the theoretical distribution.

So, in the case of R, the `dnorm`

command returns the Standard Normal distribution. That is the distribution whose probability density function is:

f(x|μ,σ)=1σ√2πe−(x−μ)22σ2

and where we **know** μ=0 and σ=1 so we actually have

f(x)=1√2πe−x22

and

P(X≤x)=∫x01√2πe−t22dt

That’s because we start knowing everything.

With the EMPIRICAL distribution we start knowing **nothing**. What we have is a collection of observations, and we want to try and derive some knowledge from that collection. Perhaps we will fit a distribution, perhaps if we have enough observations, we’ll just measure from those.

For example, if I have the following 10 numbers, I can create an empirical distribution: 1,2,3,4,4,5,8,9,9,10

Looking at just these numbers, the *empirical* probability of choosing a 5 *or less* is 60%, since I have 6 out of 10 observations of 5 or less.

What `density`

does is run through the collection of observations and fit a kernel-smoothed density to them. It isn’t normal, binomial, Poisson, Pareto, or anything in particular necessarily. It is a (sometimes) smoothed version of a histogram which can be treated like a density for calculations relating to the observations. We can try and fit theoretical distributions which are “close” in some way to the empirical. These fitted theoretical distributions can then be used as a proxy and we can use their properties for further fun and games.

**Attribution***Source : Link , Question Author : Bratt Swan , Answer Author : Avraham*