Now that we understand the idea of a probability math function, we want to talk about the special case
of the uniform probability distribution, which is where all of the values that the discrete random
variable can take on have equally likely outcomes or an equal probability of occurring.
So we looked at before this general form of a probability mass function where we said if we have a discrete
random variable capital x, the form of our probability mass function is written this way where we say
that the probability that the discrete random variable takes on the specific value x one is given here.
And then we say that this probability occurs when the discrete random variable takes on the value x
one.
And then on the second line here, we give the next possible value that X can have and its associated
probability, and we keep adding more and more lines until we run out of values x one, x two, x three,
etc. that the random variable capital x can possibly have.
So that's our general form.
When we have a discrete uniform distribution, this is what happens to that probability mass function.
All those probabilities are constant.
And so if we say that that constant probability is equal to C, so C can take on any value between zero
and one, but that constant probability is C for all of the possible values of x x sub.
I like these x1x2x3x4 values that we can see for this discrete random variable capital x, and then
the probability is zero for any other values.
In that context, one of the simplest examples of a discrete uniform distribution is the probability
distribution.
We see four dice rolls when we roll a six sided die.
So if we were to build that probability mass function, here's what we need to think about.
If we roll a six sided die one time, we know, of course, that the values we can get back are one,
two, three, four, five or six.
So we could go ahead and list those possible values one, two, three, four, five, six.
And then we need to find the probability that one of these values occurs.
Well, each of these values has a one in six chance of occurring.
We have a one in six chance of rolling a1a1 in six chance of rolling a to a one in, six chance of rolling
a three, etc. all the way up to six.
We have six equally likely outcomes.
And so for any particular outcome, the probability is one in six.
So we could write that as one in six or we could convert it to a decimal.
The decimal value is approximately 0.167.
So in our probability math function here, we're saying that the probability of getting a one is 0.167.
The probability of getting a two is 0.167.
The probability of getting a three is 0.1, six seven, etc. And these are all of the possible values,
all of the x values for the discrete random variable x, so the probability of anything else is zero.
And so we could say otherwise the probability is zero.
And this makes sense as a probability mass function because this is a discrete random variable.
We have a countable set of discrete values.
There are six values the variable can take on and they are discrete, they're countable.
There are no in-between values.
We can't measure in between two and three to get a dice roll of 2.5.
These are the only values.
So we have a discrete random variable so we can say that the random variable x is.
Discrete.
So that's our first criteria for the probability mass function.
The second criteria is that the probability of each of those values is between zero and one.
And therefore also by definition, that none of those values are negative.
And we can see that that's the case.
This 0.167 value is between zero and one, and it's not negative.
So this probability associated with each of these outcomes meets that criteria.
We can say that all of the p of x sub P values are between zero and one.
So we meet that criteria and then these values have to sum to a total of one.
And we have this 0.167 probability six times and 0.167 APR times six is equal to one.
So the probability of all of the values of X is one, and we meet that criteria as well.
Think about this another way to right this probability mass function for the discrete uniform distribution
is if we broke it out like this, we could say 0.16700.167 for x equals one.
We could say 0.167 for x equals to 0.167 for x equals three.
And we could keep going, putting each value on its own line.
We've just summarized the probability mass function.
We've consolidated it by putting everything on one line.
But this says the same thing that this does.
It's just consolidated into one line instead of six.
But we have our probability mass function.
We have it described algebraically this way now we can go ahead and graph it.
And that looks like this.
We have along the horizontal axis here all of the possible equally likely outcomes of our die role.
We can roll anywhere from a one to a six.
And then the probability that our discrete random variable takes on any of those specific values of
x is p of x.
Here, our probability for each of those values is 0.167.
So if we sketch in this graph, we see here a uniform probability across each of these equally likely
outcomes.
And that is why we call it a uniform distribution, because the probability distribution is uniform,
it stays the same no matter which value we choose.
For X, we can see that the value is level and constant everywhere along this horizontal axis for all
of these different values of X, and that value is C.
If we're talking about this form here where that probability P of X stays level or uniform at this constant
C.
So that's the simple idea behind a uniform distribution.
But as this die roll example indicates, we see it all the time in the real world.
So it's important for us to understand that when we have uniform probability across the full set of
outcomes, the result is this discrete uniform distribution, the uniform distribution of the discrete
random variable.