Question

http://www.eng.usf.edu/~reeves/quiz.html
Hi guys I am not asking you to do the quiz for me, but a simple explanation of how to differentiate the types of distributions.

Answer 1

There is: Poisson, Binomial, Negative Binomial, Normal, Hyper-geometric, Exponential, and Lognormal.

Answer 2

Is my professor asking for too much? or should there be a simple way to remember each of the different distributions.

Answer 3

I know Discrete is: Binomial, Negative Binomial, Hyper-geometric, and Poisson.

Answer 4

I have done this countless times, but do not understand at all why this could be a poisson distribution, when it clearly states that it is exponential.

Answer 5

Is this a mistake on my professor's part? Or is it really a poisson distribution?

Answer 6

I think these distributions are standard to know in a probability or stats course. So, I don't find it unreasonable to know. It can be a bit confusing at first, but let me break it down:

\(\textbf{Hypergeometric}\):
This situation occurs when you are sampling WITHOUT replacement, and you are sampling from a population that can be described in two categories ("success/failure"). As you are making selections from the population, each draw decreases the population, implying the the probability of success changes with EACH draw. If \(X\) is the number of successes, then \(X\) is said to have a hypergeometric distribution.
\(\textit{Example}\): You have 30 balls in an urn. 20 of them are red, 10 of them are green. (Your population is divided in 2 categories: red and green balls). You draw a sample of 5 balls without replacement. Now, if \(X\) is the number of red balls, the probability of drawing "1 red" for example, changes on each draw since you are removing balls from the population. (First draw: P(1 red ball) = 20/30 , Second draw: P(1 red ball)=19/29 since one red ball was removed on the 1st draw, so the probability of the success (one red ball) changes from draw to draw.).

\(\textbf{Binomial}\):
This situation occurs when you are sampling WITH replacement, and you are, again, sampling from a population that can be described in two categories ("success/failure"). Each trial is INDEPENDENT of each other. So, as you are making selections from the population, the probability of success is the SAME on each draw. If \(X\) is the number of successes obtained, then \(X\) is said to have a binomial distribution distribution.
\(\textit{Example}\): Consider the ball example above, but instead you sample WITH replacement. So it means, that after every draw, you put back the ball you took on a draw. So, the probability of success, P(1 red ball) = 20/30 on all 5 draws.

Note: If you have a hypergeometric situation (without replacement), but are said to have a very large population (and you sample a small number from the population), then you can approximate the hypergeometric with a binomial (because the probability of each success, although they change on each trial, don't change MUCH if the population is very large).

\(\textbf{Negative Binomial}\):
Is kind of similar to the binomial. The population can be described again in two categories (success/failure). Here, you have repeated trials, and each trial is INDEPENDENT of each other. So, each success probability is the SAME on each trial. You repeat the trials until you obtain a certain number of successes have been obtained. Then, if \(X\) is the number of FAILURES before a certain number of successes, then \(X|) has a negative binomial distribution.
\(\textit{Example}\): Consider the ball and urn example from above, in the case with replacement. But, if you are asked to find the probability of getting 3 blue balls (3 failures) before the 4th red ball (4th success), then if \(X\) is the number of blue balls, then \(X\) has a negative binomial distribution.

Note: You can also let \(X\) be the number of TRIALS to get a certain number of successes. It's still negative binomials, but the probability mass function looks a bit different, and has a different support.

A quick comparison between Binomial and Negative binomial:
-\(text{Binomial}\):The number of trials are given, the random variable is the number of successes, and the support is finite.
-\text{Negative binomial}\): The number of successes is given, the random variable is the number of trials (or failures), and the support is infinite.

\(\textbf{Poisson}\):
Doesn't really look relatable to the other 3 distributions above in a practical sense (although technically, the Poisson distribution does arise as the limiting distribution of the binomial as \(n \to \infty\) and \(p \to 0\)). If \(X\) represents the number of events occurring in a fixed time interval (or "space"), then \(X\) has a Poisson distribution if these events occur at some average rate and and if the number of occurrences in non-overlapping time intervals is independent. In some examples, they mention if the number events follow a Poisson distribution.
\(\textit{Example}\): Say you are working at a call centre. Then if \(X|) is the number of phone calls received at the centre in an hour, then \(X\) has a Poisson distribution.

\(\text{Exponential, Normal, Lognormal\):
These are harder to identify if the example doesn't tell you the distribution. . You can't really tell unless what you have unless they explicitly tell you what the distribution is (unless you know that certain physical applications or phenomena occur according to some of these distributions).
But, for the \(textbf{Exponential}\) distribution, it's often used to model the time until an event occurs.
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As for the example you provided up there^ , there is a close link between the exponential distribution and the Poisson distribution. Suppose \(X\) has an exponential distribution. Then we can regard \(X\) as being the \(\textbf{time}\) between successive occurrences. Now let \(T\) be the \(\textbf{number}\) of events that occur in a time interval, then \(T\) will have a Poisson distribution. So, you can basically think of it as: the exponential distribution describes the time between events, and the Poisson distribution describes the number of events occurring in a time interval. It's important to distinguish is they are asking about time

So in your question, yes the days between airplane accidents has an exponential distribution, but the NUMBER of events ( accidents) will have a Poisson distribution.

Just as a note: for the Poisson distribution, it doesn't just have to describe the number of events in a time interval, it can also describe the number of events that occur in "space", such as the number of events occurring over a certain length, area, or volume.

Answer 7

urgh I'm sorry for the formatting mistakes

Answer 8

Thank you so much, this is a great detailed explanation, I can read eveyrthing just fine!

Answer 9

But one thing, my professor strangely has hyper-geometric and geometric as two different distribution types. Which I can distinguish the geometric part?

Answer 10

Geometric is actually related to the Negative binomial. In fact, the Negative binomial is often to be X is "The number of failures before the \(r\)-th success. The geometric distribution is the a SPECIFIC case of the negative binomial when \(r=1\), that is X is |"The number of failures before the 1st success"

Answer 11

So to relate to the ball example... if you are asked to find the probability of getting 3 blue balls (3 failures) before the 1st red ball (1st success, r=1), then if X is the number of blue balls, then X has a negative binomial distribution.

Answer 12

oh that makes perfect sense! geometric and negative binomial. I don't understand why they are listed differently, maybe to throw students off.

Answer 13

sorry, X has a geometric distribution* in the last sentence (but technically it also has a negative binomial distrbution too..)

Answer 14

I am not sure either, but I guess if you are lucky to encounter the geometric problem, you don't have to recall the "complicated" probability mass function of the negative binomial

Answer 15

*Fifty percent of the exam will be comprised of questions for which you will have to complete the following: a) indicate if the random variable is discrete or continuous, b) indicate the correct distribution associated with the random variable (e.g. normal, exponential, binomial, etc.), and c) write the value of each parameter necessary for the distribution. YOU WILL NOT ACTUALLY SOLVE THE PROBLEM.*

Answer 16

discrete or continuous is easy, i just have to get the hang of indicating the correct distribution, and assigning values to the parameter necessary which I think is a lot to remember sometimes. But thank you for your help honestly!

Answer 17

yeah it is a bit confusing at first... When I first encountered these in my first course I was quite overwhelmed and kind of struggled. But with time I definitely got the hang of them by doing manyyy problems, and also understood where the probability mass functions came from (I sort of just memorized them at first without understanding them).

Answer 18

I;m gonna be honest that's kind of what I am doing right now, but thank you for your encouraging words. I have an exam for Stats for Engineers monday and a Calc III exam tuesday.

Answer 19

I'm sure you'll get through with practice :) I thought I hated probability before it was confusing, but now I actually really like it xD Good luck with your exams!!

Answer 20

Thank you for your help and kind words, I will try my best! You have really cleared the gray clouds for me and revealed sun shine, thanks again!

Answer 21

oh you're welcome :)