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### Course: Statistics and probability>Unit 9

Lesson 9: Poisson distribution

# Poisson process 1

Introduction to Poisson Processes and the Poisson Distribution. Created by Sal Khan.

## Want to join the conversation?

• seriously, just got lost when there was intro of limits and the evaluation of e. someone please explain what does that has to do with the Poisson process, and if possible in simple terms what really is Poisson process and its relevance?
• The normal distribution has a bell curve as its probability distribution. Many practical real-world measurements follow a bell curve. Averages of samples from any distribution can be approximated with a bell curve by the Central Limit Theorem. The Central Limit Theorem is very powerful, but there are certain limitations, and assumptions must be made to use it. It is used often in sampling theory and hypothesis testing.

The Poisson and Binomial distributions are discrete "counting" distributions. The regular probability distributions (not sampling distributions) are generally skewed, not symmetric like the normal distribution (which by the way is continuous, not discrete). The Binomial distribution assumes a predetermined number of trials, but the Poisson has no upper limit of possible successes. This is why limits are used to show the relationship between the two distributions.
• What are the prerequisites to studying more advanced statistics? I'm interested in learning statistics but I haven't done much study of calculus or any of the other higher mathematics.
• ^ I will correct you here. You will need to have done a first-year math course in calculus and algebra. a second-year calculus course is highly recommended. most universities now require you to have done a basic probability course, normally offered at second year. Mathematical statistics is a very hard course to take if you don't understand basic probability, calculus and algebra.
• How is lambda/60 a probability?
• lambda = n.p where n is number of intervals and p is probability of a success.
Divide both sides by n
lambda/n = p
• On , Sal says that P(X=k) is the probability of k cars passing in an hour. On , he says, "What happens if one car passes in an hour? Or more than one car passes in a minute? " These two statements are confused me.
• "in and hour" was a mis-statement that Sal corrects immediately by restating "in a minute".
• why DOES Poisson Dist., Binomial Dist. , and Normal Dist. act as a BELL CURVE? is it really that fascinating or there is some specific reason behind it?
• To go a bit further than the above answer, Poisson, Binomial, and Normal are all related (and all have continuous analogs, which are likewise related). Poisson is a special case of binomial in which n (the number of events) is very high and p (the probability of each event) is very low.

While you should understand the proof of this in order to use the relationship, know that there are times you can use the binomial in place of the poisson, but the numbers can be very hard to deal with. As an example, try calculating a binomial distribution with p = .00001 and n = 2500. Mind you, this will require you to do 2500!, which is not very convenient. On the other hand, converting it into a Poisson problem makes it much more manageable.

The normal distribution on the other hand can be used with any sample mean and the Central Limit Theorem. It's all part of the awesome cycle of life. :)
• I believe the answer to the calculation in the end is 0.004998097 or 0.5% (we want to check “2” with lambda 9).
If I try 9 with lambda 9, I get 13%. Isn’t this value too low assuming the expected value IS 9?
• Good question! It's easy to think that the probability of the mean value should be higher than 13%, but when there are lots and lots of possible values, then even the most likely value won't have a very high probability. For example, if I flip a coin 2 times, and I get exactly 1 heads, that's no surprise. But if I flip a coin 1000 times and get EXACTLY 500 heads, that's actually pretty amazing, even though 500 is definitely the most likely outcome (and the mean). So in this example, if you calculate the probabilities for every value from k = 0 to k = 20, you will see that the distribution DOES peak around k = 9, but it's just spread out a bit. The values near k = 9 are:
6 9.1%
7 11.7%
8 13.2%
9 13.2%
10 11.9%
11 9.7%
12 7.3%
So you see that if you have to divide up 100% into lots of possibilities, even the most likely one might not actually be very likely!
• I am so lost, I don't understand anything at all
• The easiest way to do this is:
(e to the negative Lambda) multiplied by (x to the k) / divided by k factorial.
So, (e^-L)(L^k)/k!
It is really simples because there is a lot of repetition in the setup. e to the - Lambda, Lambda to the k, over k!; e, Lambda, Lambda, k, k
• How can we have probability distribution with parameter as time as Binomial Distribution?
Is it because we have two answer yes or NO for a car passing to the min or not?
IF yes then is it that we can model everything in the universe with Binomial distribution?
• The answer is "it depends". The Binomial Distribution is a very powerful and versatile tool in statistics, but it doesn't cover everything. There are a number of conditions that the binomial depends on, such as independence, that does not apply in all situations.

I guess a simple way of putting it is that a lot of distributions can be thought of as "like the Binomial, but for Special Case X".

I'm oversimplifying a lot, but there are a lot of cases in statistics where a distribution is derived from another distribution for specific cases, a classic example being the Normal distribution and Student's t. Yes, in large samples, the Normal and t distributions are identical, but what if you don't have a large sample? Then t works better. The same logic applies to the Binomial, the Poisson, and the whole family of distributions derived from them.

The field of density functions in statistics is vast and varied, and Khan Academy, and indeed, any intro to statistics course you may take in college, will only be able to scratch the surface of this topic.

However, statistics is easier than you think, if you focus more on the utility and less on the math. In this age of technology, nobody will care if you can calculate a density function. Any computer can do that in milliseconds. What is far more important is if you can explain what the function means, and how it is important. That's why I always tell my students that statistics is much more of a writing class than a math class. So don't intimidated by the calculations, and focus on understanding of the general idea. Good luck!
(1 vote)
• Is there a specific video for e, or for the limit approching infinity, that Sal mentions at ?