Uniform Distribution EXPLAINED with Examples 

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Thank you for the positive feedback! Yea, intro to stat courses often don't do a good job at explaining where certain things come from. If you are interested, here is a derivation of the uniform distribution standard deviation formula: www.quora.com/What-is-the-standard-deviation-of-a-uniform-distribution-How-is-this-formula-determined I'm glad we were able to help out!

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Great idea! In short, the entropy of a discrete distribution is defined as "how much uncertainty" it has. If a certain outcome is 100% certain and the other outcomes have a probability of 0, then the entropy of the distribution is 0 because there is no uncertainty in what the outcome will be. On the other hand, the entropy is maximized when all probabilities for the outcomes within the distribution's interval are the same (AKA the uniform distribution). This resource does a good job explaining it: tdhopper.com/blog/entropy-of-a-discrete-probability-distribution but we will add this topic to our list of future videos!

glad I'm not doing what you are doing holy shit.

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Thank you a lot of watching! :) Yes, that is something we plan on tackling in the future.

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Thank you for your kind words! Yes, we plan on tackling topics like this in the future!

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If you look through our channel you will see various videos on these topics.

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It's definitely something I believe and know some students can lose sight of when struggling with a class. Thanks for watching!

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Thank you! We use Manim to generate the animations!

Great question! We got 10 and 2 from the problem statement that said arrives uniformly late between 2 and 10 minutes.

We are working on putting out some new videos soon. Thanks for the comment!

We are using software called Manim to make the animations. It's free and open-source too!

Hey Elizabeth, great questions! Yes, the area is the space under the curve between whichever x values you are trying to find the probability of. As for the standard deviation, the derivation of that formula is a little complex and outside the scope of this video. It comes from some statistics identities that are applied to the uniform distribution. As a result of performing those calculations, a 12 happens to come out. A great derivation and explanation of this standard deviation formula can be seen here: www.quora.com/What-is-the-standard-deviation-of-a-uniform-distribution-How-is-this-formula-determined As for the mean (mu), yes, you are always going to divide by 2. This is because the mean happens to be directly in the middle of curve due to its symmetry. So essentially you are finding the middle or average of the bounds a and b. To take the average of 2 numbers, you add them up and divide by two. Let me know if you have any additional questions!

@@AceTutors1 thank you! My college textbook gave us a formula and I had no idea how to use it and my professor didn't explain it well so I'm two weeks behind the class and have been binge watching anything off RU-vid but this was the first video that made sense!!! Thank you so much! And thanks for the link!

@@lk6977 I love to hear that we were able to help you out! We have a bunch more Stat videos on similar topics so be sure to check those out too! We appreciate your message! :)

Great question! That is a very tricky concept, but it ultimately comes from the fact that the uniform distribution is a continuous distribution rather than a discrete distribution. Because it is continuous, the probability of x equaling any specific number is actually 0! For continuous distributions, we can actually only calculate probabilities across an interval of numbers like from 7-10 or 3-4 or something. With this information, we actually find out that the probability of being greater than 7 is the same and being greater than or equal to 7 because that additional probability of being equal to 7 is actually 0, so it doesn't add anything. I know this is a bit of a complex concept to wrap your head around, so if you have questions about this, please feel free to reach out!

Nice! I'm glad we were able to help!

That is something we definitely plan on covering soon! Thanks!

Great question! The F-distribution and normal distribution are similar in that they are used in hypothesis testing and confidence intervals, but they are used in different contexts. The F-dstribution is often used in ANOVA testing and to test if variances of two populations are equal.

To answer that question, we would need a bit more information. From that info, we know that the probability (or area) of being to the right of 1/3 is 0.5. Since this distribution is uniform, the fact that we have the area is 0.5 tells us that 1/3 must be exactly halfway between the endpoints a and b. However, we don't know exactly how spread out the distribution is. For example, if a = 0, then we could figure out that b must be 2/3 in order to satisfy the fact that 1/3 is directly in the middle. However, if a = 1/6, then this would imply that b is 3/6 (or 1/2) in order to satisfy 1/3 being in the middle. Without any additional information we wouldn't be able to say exactly what a and b are other than that they are equidistant from 1/3.

We actually do have a video on this topic. You can find it here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-xI9ZHGOSaCg.html&lc=UgzDaA0mYL22gzIo4VJ4AaABAg

Great question! You are right that the probability distribution has the same height or value throughout the entire interval for a to b, but probability itself is actually the area under the curve. The area (AKA the probability) is different from a to b compared to c to d.

That means that the time that the bus is late is uniformly distributed. This means that there is an equal chance (or probability) of the bus being late anywhere between 2 and 10 minutes. That means it has an equal chance of being 3 minutes late vs 5 minutes late vs 6.397 minutes late. This can visualized by the the probability distribution just being a straight horizontal line.

Great question! And that is actually beyond my area of expertise, unfortunately. Perhaps a resource like this can be helpful: dlsun.github.io/probability/autocorrelation.html

Hi Russel! To solve for the standard deviation, you would use the formula discussed in this video, where b and a are the upper and lower limits of the distribution.

Great question! You can find the percentile of a given point within the distribution directly by finding the proportion your point is from the beginning (left side) of the distribution's interval with respect to the entire interval's width. For example, if its uniformly distributed from a=1 to b=5 and we want to find the percentile of some point c=2, we would divide (c-a)/(b-a) which would be (2-1)/(5-1) = 1/4 = 0.25 which means c would be the 25th percentile. I hope this helps! :)

Hi Dennis, great question! So the answer to this is the difference between discrete and continuous distributions. If it were a discrete distribution, you'd be right. The only outcomes would be all whole numbers between 2 and 10. However, this is actually a continuous distribution. This means the sample space is actually all the whole numbers you mentioned, as well as everything in between. It's infinite! The bus could arrive exactly 7 minutes late, but it could also arrive 7.5 minutes late, 7.25 minutes late, 7.00000001 minutes late and so on. So instead of taking the # of possibilities that satisfy what you're looking for and dividing by the total # of outcomes, as you would if this uniform distribution was discrete, you would actually take the width of the desired sample space (7 to 10) and divide by the total width (2 to 10). This results in the fraction 3/8 we found in the video. If you have any other questions, please feel free to ask!

@@AceTutors1 Thank you! Very helpful and clear video explanation

@@dennisdwitama9206 Thank you Dennis for the support! I'm glad we were able to help!

Yes it is! Great question!

Great question Ariah! In order to understand that, you need to know whether this is a discrete or continuous distribution. The uniform distribution is actually continuous! This is because the possible outcomes for the arrival of the bus is infinite between 2 and 10. The bus could arrive exactly 7 minutes late, but it could also arrive 7.5 minutes late, 7.25 minutes late, 7.00000001 minutes late and so on. Because there are an infinite number of outcomes, the probability of any one of those outcomes is always actually 0! That is why, for continuous distributions like this one, we will only ever really be asked to find the probability between 2 values like we did in this video. I hope this helps!

@@AceTutors1 thank you! Well said, and exactly what I need!

That's something we plan to cover soon! :)

Great question! So the mean is (a+b)/2 because you are essentially finding the midway point between a and b or the average of two number which is what the formula gives you. (b-a)/2 would give you half of the interval between a and b or the distance that the mean is from either endpoint, so you could take that value and add it to a or subtract it from b to get the mean.

Hahah that's great. You can check out our website: theacetutors.com to see if we are at a university near you! ;)

So the same y value existing for all x values between a and b can be seen by the flat line in between those points. At every x between a and b, the y value is the exact same, so when you plot it, the line looks horizontal; it does not go up or down at all.

That's a topic we plan to cover soon! Thank you for your suggestion!

Hahah thanks!

Thanks for your comment! Yea, this video is just meant to be a high level overview of the continuous uniform distribution and its overall key formulas.

Thanks for the great feedback! We will consider making videos for these topics in the future

You're beyond welcome!