What is an even distribution in statistics

equal distribution

Here you can find out everything Uniform distributions. First the discrete uniform distribution is dealt with, then the continuous uniform distribution. Among other things, the Density function, the Distribution function, the Expected value and the Variance calculated for the discrete and continuous case of the probability distribution using an illustrative example.

Would you rather understand everything right away without reading this article? Then our videos are for discrete uniform distribution and to constant uniform distribution just the thing for you!

Equal distribution simply explained

The equal distribution is one Probability distribution of statistics. It will be between the discrete uniform distribution and the constant uniform distribution differentiated. In the steady case, this distribution will also be Uniform distributioncalled. Fundamentally, the two differ in that in the discrete case all possible outcomes have the same probability and in the continuous case the density is constant.

Discrete equal distribution

The discrete uniform distribution is one of the simplest distributions in probability. It is present when a random variable is discrete, i.e. it only has a finite number of possible outcomes and each outcome has the same probability.

For now, however, we will limit ourselves to one Example of a discrete uniform distribution, namely random experiments, the possible results of which can be represented by integers between a and b.

A classic example of use is the throwing of a dice, which of course is not marked. The probabilities are evenly distributed here. Evenly distributed means that in this example every possible result between a equal to 1 and b equal to 6 occurs with the same probability.

Equal distribution probability

Let us start with the probabilities of the discrete uniform distribution and the associated probability function, which is the “density function” for the discrete case. When rolling the dice, every result has a probability . The function then looks like this:

The formula might look a bit complicated, but it's actually quite simple. For every evenly distributed result between 1 and 6, the probability is the same . Since nothing else is possible when rolling the dice, the probability of other outcomes is 0.

Equal distribution Distribution function: discrete

In the general case, the distribution function looks a bit strange:

The two straight lines |… | stand for the power of the crowd. So let's look for the probability of a result, for example , we count all possible outcomes that are less than or equal to 4, i.e. 1,2,3 and 4 when rolling the dice. That means our quantity in the numerator has 4 elements. So the following applies:

As you know, the distribution function always indicates the probability that a result that is less than or equal to x will come out. The first section applies to results less than a, for example the result 0 when rolling the dice. Since it is not possible to get this result, the probability is equal to 0. The second section applies to results between a and b, i.e. in ours Case between 1 and 6. [x] stands for the rounding of x. The distribution function of the example of the discrete uniform distribution is therefore also divided into three parts:

Expected value uniform distribution: discrete

In this case, the expected value of the discrete uniform distribution is simply the mean of a and b, i.e. a plus b divided by 2.

In the general case, this formula applies to the expected value:

Variance Equal distribution: discrete

The formula of the variance in the case treated here is as follows:

This formula applies to the variance of the discrete uniform distribution in the general case:

Here you will find all the important formulas again, for the example considered:

In general, the formulas for discrete uniform distribution are as follows:

Constant equal distribution

The constant uniform distribution is a probability distribution that has a constant probability density over an interval. Every conceivable real value of the random variable is equally likely in a given interval. This is where the name comes from uniform distribution.

That doesn't seem quite understandable to you yet?

Then imagine the whole thing with an example. Let's say it's Saturday night and you're on your way home from the club.

You know that the S-Bahn only runs every hour at night, but you forgot the exact departure times. So if you run to the station with good luck, your waiting time is a constant even distribution between a equal to zero and b equal to sixty. Because all times are between zero and sixty minutes uniformly distributed. That means you can wait any time you can, for example 5.2343 minutes. Isn't it logical?

In short, it looks like this:

or in general

Expected value uniform distribution: continuous

You can calculate the expected value in the continuous case with the following formula:

You can see that the expected value is exactly in the middle of a and b.

Variance equal distribution: continuous

You can calculate the variance of the constant uniform distribution with this formula:

Don't worry, we'll save you the math here. The best thing to do is to memorize these formulas or write them down on your formula sheet.

Density function uniform distribution

You represent the density function of the constant uniform distribution as follows:

The density function can be roughly divided into two parts. Within the observed interval, all have values ​​- here too carrier called - the same probability. This is expressed with. Outside this range, the probability is always 0. This also explains the two-part definition of the density function of the continuous uniform distribution.

Uniform distribution Distribution function: continuous

The associated distribution function is defined in three parts:

This can also be easily explained if you look at the graph. Within the observed interval, the distribution function is a straight line that rises constantly from 0 to 1. This is because the cumulative probabilities are evenly distributed. At the point x = a, the function is equal to 0 and continuously approaches the value 1 as it approaches b.

Constant equal distribution example

Let us return to our consideration from above. You're dead tired on your way to the S-Bahn station. Since you want to get home to bed as soon as possible and you know exactly that if you wait more than 15 minutes at the platform you will fall asleep, you calculate how likely it is that you will have to wait less than 15 minutes.

To do this, you use the formula of the distribution function and insert our values.

So the probability that you will have to wait 15 minutes or less is 25 percent. It's a shame, so you'll probably be spending the night on the platform.

But have fun aside! You can now calculate the expected value and the variance yourself by inserting the values ​​into the formulas. We will give you the solutions:

That's it for the equal distribution! You now know how to calculate it and that you should leave the club earlier next time.