Introduction to Discrete Uniform Distribution

Introduction to Discrete Uniform Distribution


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.

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