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