Welcome back!
In this video we will talk about discrete distributions and their characteristics.
Let’s get started!
Earlier in the course we mentioned that events with discrete distributions have finitely
many distinct outcomes.
Therefore, we can express the entire probability distribution with either a table, a graph
or a formula.
To do so we need to ensure that every unique outcome has a probability assigned to it.
Imagine you are playing darts.
Each distinct outcome has some probability assigned to it based on how big its associated
interval is.
Since we have finitely many possible outcomes, we are dealing with a discrete distribution.
Great!
In probability, we are often more interested in the likelihood of an interval than of an
individual value.
With discrete distributions, we can simply add up the probabilities for all the values
that fall within that range.
Recall the example where we drew a card 20 times.
Suppose we want to know the probability of drawing 3 spades or fewer.
We would first calculate the probability of getting 0, 1, 2 or 3 spades and then add them
up to find the probability of drawing 3 spades or fewer.
One peculiarity of discrete events is that the “The probability of Y being less than
or equal to y equals the probability of Y being less than y plus 1”.
In our last example, that would mean getting 3 spades or fewer is the same as getting fewer
than 4 spades.
Alright!
Now that you have an idea about discrete distributions, we can start exploring each type in more detail.
In the next video we are going to examine the Uniform Distribution.
Thanks for watching!
4.4 Uniform Distribution
Hey, there!
In this lecture we are going to discuss the uniform distribution.
For starters, we use the letter U to define a uniform distribution, followed by the range
of the values in the dataset.
Therefore, we read the following statement as “Variable “X” follows a discrete
uniform distribution ranging from 3 to 7”.
Events which follow the uniform distribution, are ones where all outcomes have equal probability.
One such event is rolling a single standard six-sided die.
When we roll a standard 6-sided die, we have equal chance of getting any value from 1 to
6.
The graph of the probability distribution would have 6 equally tall bars, all reaching
up to one sixth.
Many events in gambling provide such odds, where each individual outcome is equally likely.
Not only that, but many everyday situations follow the Uniform distribution.
If your friend offers you 3 identical chocolate bars, the probabilities assigned to you choosing
one of them also follow the Uniform distribution.
One big drawback of uniform distributions is that the expected value provides us no
relevant information.
Because all outcomes have the same probability, the expected value, which is 3.5, brings no
predictive power.
We can still apply the formulas from earlier and get a mean of 3.5 and a variance of 105
over 36.
These values, however, are completely uninterpretable and there is no real intuition behind what
they mean.
The main takeaway is that when an event is following the Uniform distribution, each outcome
is equally likely.
Therefore, both the mean and the variance are uninterpretable and possess no predictive
power whatsoever.
Okay!
Sadly, the Uniform is not the only discrete distribution, for which we cannot construct
useful prediction intervals.
In the next video we will introduce the Bernoulli Distribution.