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“Again in this lecture. We are going to talk about nvarious types of probability distributions distributions and what kind of events. They can be used to ndescribe certain distributions share. So we ngroup them into types.
Some like rolling a die or picking a card nhave. A finite number of outcomes. They follow. Discrete distributions.
Others like recording time and distance in ntrack field have infinitely. Many outcomes they follow continuous distributions. We are going to examine the characteristics nof. Some of the most common distributions for each one.
We will focus on an important naspect of it or when it is used before we get into the specifics you need nto know the proper notation. We implement when defining distributions we start off by writing down the variable nname for our set of values followed by the tilde sign. This is superseded by a capital letter depicting nthe type of the distribution and some characteristics of the dataset in parenthesis. The characteristics are usually mean and nvariance.
But they may vary depending on the type of the distribution alright let us start by talking about the discrete nones..
We will get an overview of them and then we nwill devote a separate lecture to each one so we looked at problems relating to drawing ncards from a deck or flipping. A coin. Both examples show events where all outcomes nare equally likely such outcomes are called equiprobable and nthese sorts of events. Follow a discrete uniform distribution.
Then there are events with only two possible noutcomes true or false. They follow a bernoulli distribution regardless nof whether one outcome is more likely to occur any event with two outcomes can be transformed ninto. A bernoulli event. We simply assign one of them to be true nand the.
Other one to be false imagine. We are required to elect a captain nfor our college sports team. The team consists of 7 native students and n3 international students. We assign the captain being domestic to be n true and the captain being an international as false since the outcome can now only be true nor false.
. We have a bernoulli distribution. Now if we carry out a similar experiment nseveral times in a row. We are dealing with a binomial distribution.
Just like the bernoulli distribution..
The noutcomes for each iteration are two. But we have many iterations for example. We could be flipping. The coin nwe mentioned earlier 3 times and trying to calculate the likelihood of getting heads ntwice lastly.
We should mention the poisson distribution. We use it when we want to test out how unusual nan event frequency is for a given interval for example imagine we know that so far lebron njames averages 35 points per game during the regular season. We want to know how likely it is that he will nscore 12 points in the first quarter of his next game. Since the frequency changes so should our nexpectations for the outcome using the poisson distribution.
We are able nto determine the chance of lebron scoring exactly 12 points for the adjusted time interval great now on to the continuous distributions one thing to remember is that since we are ndealing with continuous outcomes. The probability distribution would be a curve as opposed to nunconnected individual bars. The first one we will talk about is the normal ndistribution the outcomes of many events in nature closely nresemble this distribution hence the name normal for instance. According to numerous reports nthroughout the last few decades the weight of an adult male polar bear is usually around n500 kilograms.
However there have been records of individual nspecies weighing anywhere between 350kg and 700kg extreme values like 350 and 700 are called noutliers and do not feature very frequently in normal distributions. Sometimes we have limited data for events nthat resemble a normal distribution in those cases. We observe the student s t ndistribution. It serves as a small sample approximation nof.
A normal distribution..
Another difference is that the student s t. Naccommodates extreme values significantly better. Graphically that. Is represented by the curve nhaving.
Fatter. Tails . Overall this results. In more values.
Extremely nfar away from the mean. So. The curve would probably more closely resemble a student s t ndistribution than a normal distribution. Now imagine.
Only looking at the recorded weights nof the last 10 sightings across alaska and canada. The lower number of elements would make the noccurrence of any extreme value represent a much bigger part of the population than nit should good job. Everyone another continuous distribution. We would like nto introduce is the chi squared distribution.
It is the first asymmetric continuous distribution nwe are dealing with as it only consists of non negative values graphically that means that the chi squared ndistribution..
Always starts from 0 on the left depending on the average and maximum values nwithin the set the curve of the chi squared graph is usually skewed to the left unlike the previous two distributions the nchi squared does not often mirror real life events. However. It is often used in hypothesis testing nto help determine goodness of fit the next distribution on our list is the exponential ndistribution. The exponential distribution is usually present nwhen.
We are dealing with events that are rapidly changing early on an easy to understand example is how online nnews articles generates hits they get most of their clicks. When the topic nis still fresh the more time passes. The more irrelevant nit becomes and interest dies off the last continuous distribution. We will mention nis.
The logistic distribution. We often find it useful in forecast analysis nwhen. We try to determine a cut off point for a successful outcome for instance take a competitive e sport like ndota. 2.
We can use a logistic distribution to determine nhow much of an in game advantage at the 10 minute mark is necessary to confidently predict victory nfor either team ” ..
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