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Overview of Some Discrete Probability Distributions (Binomial,Geometric,Hypergeometric,Poisson,NegB) - YouTube
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Let's look at a quick overview of some
discrete probability distributions and their relationships.
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I intend this video to be used as a recap
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after having been introduced to these distributions,
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but it could possibly be used as an
introductory overview.
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I don't any calculations in this video,
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nor do I discuss how to calculate the probabilities.
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I simply discuss how these different distributions arise,
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and the relationships between them.
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The Bernoulli distribution is the distribution
of the number of successes on a single Bernoulli trial.
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In a Bernoulli trial we get either a success or a failure.
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It's like an answer to a yes or no question.
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A Bernoulli random variable can take
on only the values 0 and 1.
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For example, we can use the Bernoulli
distribution to answer questions like:
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if a single coin is tossed once, what is
the probability it comes up heads?
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Or, if a single adult American is
randomly selected,
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what is the probability they
are a heart surgeon?
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Some other important distributions are built
on the notion of independent Bernoulli trials,
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where we have a series of trials,
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and each one results in a
success or a failure.
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An important one is the binomial distribution,
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which is the distribution of the number
of successes in n independent Bernoulli trials.
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For example, with the binomial distribution
we can answer a question like:
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if a coin is tossed 20 times, what is the
probability heads comes up exactly 14 times?
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And since the binomial distribution
is the distribution of
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the number of successes in
n independent Bernoulli trials,
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the Bernoulli distribution is a special case of the
binomial distribution with n=1, a single trial.
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Continuing on with the theme
of independent Bernoulli trials,
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the geometric distribution is the distribution of the
number of trials needed to get the first success.
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For example, with the geometric
distribution we can answer a question like:
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if a coin has repeatedly tossed,
what is the probability
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the first time heads appears
occurs on the 8 toss?
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The negative binomial distribution is a
generalization of the geometric distribution.
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The negative binomial distribution is
the distribution of the number of trials
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needed to get a certain number of successes
in repeated independent Bernoulli trials.
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So the negative binomial distribution can
help us answer questions like:
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if a coin has repeatedly tossed,
what is the probability
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the third time heads appears
occurs on the ninth trial?
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The way the binomial distribution and
the negative binomial distribution arise
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can sound similar, and they can sometimes be confused.
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They differ in what the random variable is.
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In the binomial distribution,
the number of trials is fixed,
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and the number of successes
is a random variable.
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For instance, we're tossing a coin
a fixed number of times,
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and the number of heads that comes
up is a random variable.
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In the negative binomial distribution,
the number of successes is fixed,
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and the number of trials required to get
that number of successes is the random variable.
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For instance, we might be tossing a coin
until we get heads 4 times.
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And the number of tosses required to get heads
4 times is the random variable.
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Now I'll talk about two distributions
that are related to the binomial,
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but aren't based on independent
Bernoulli trials.
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The hypergeometric distribution is similar
to the binomial distribution
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in that we're interested in the number of
successes in n trials,
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but it's different because the trials
are not independent.
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The hypergeometric distribution is the
distribution of the number of successes
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when we are drawing without replacement
from a source that contains
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a certain number of successes
and a certain number of failures.
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For example, we can use the hypergeometric
distribution to answer a question like:
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if 5 cards are drawn without
replacement from a well shuffled deck,
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what is the probability exactly 3
hearts are drawn?
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It's different from the binomial because
the probability of success,
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the probability of getting a heart, would
change from card to card,
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depending on what happened before.
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However, if the cards are drawn with replacement,
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meaning the card was put back in and
reshuffled before the next card was drawn,
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then the trials would be independent and we
would use the binomial distribution instead.
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If we are sampling only a small fraction of objects
without replacement from a large population
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then the trials are still not independent, but
that dependency has only a small effect,
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and the binomial distribution closely
approximates the hypergeometric distribution.
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So there are times when a problem is in
its nature a hypergeometric problem,
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but we use the binomial distribution as
an approximation.
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This can make our life a little bit
easier sometimes.
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Another distribution related to the
binomial is the Poisson distribution.
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But this one's a little harder to explain.
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The Poisson distribution is the distribution
of the number of events
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in a given time or length, or area,
or volume etc.,
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if those events are occurring randomly
and independently.
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There's a bit more to it than that, and I go into
this in much greater detail in my Poisson videos.
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But that's the gist of it. So we might
use the Poisson distribution to answer a question like:
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what is the probability there will be
exactly 4 car accidents
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on a certain university campus in a given week?
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There is a relationship between the Poisson
distribution and the binomial distribution.
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The Poisson distribution closely
approximates the binomial distribution
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if n, the number of trials, in the
binomial, is large
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and p, the probability
of success, is very small.
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So suppose we have a question like:
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what is the probability that in
a random sample 100,000 births,
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there is at least one case of progeria?
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Progeria is an extremely rare disease
that causes premature aging,
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and it occurs in about 1 in every eight
million births.
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This is truly a binomial problem.
But we have a binomial problem
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with a very large n, 100,000,
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and a very small probability of
success, 1 in eight million or so,
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because progeria such a rare disease.
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And so this could be very well
approximated by the Poisson distribution.
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I look into all of these concepts discussed
in this video in greater detail
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in the videos for these specific distributions.
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