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