Expected Values for Continuous Variables!!!

Рет қаралды 81,398

Күн бұрын

If you ever muck around in statistics, it's not long before you see E(x) = something. These are expected values. Expected Values for Continuous Variables are a little trickier than their discrete counterparts because we have to do some calculus. However, I'll walk you through it, one step at a time, so don't sweat it! BAM!
NOTE: This one is, believe it or not, pretty near to my heart. When I was taking Statistical Theory in graduate school (from Rodger Berger of Casella and Berger, who wrote the standard textbook on statistical theory, "Statistical Inference") I remember having a lot of trouble with expected values. They intimidated me for two reasons 1) deriving them seemed like total luck and 2) I never understood, exactly, what the formula for continuous variables meant in a deep way. I could look at the equation and name the parts, but that was all I could do. Anyway, fast forward a few years and here I am, going back to these basics, this time determined to get that "deep understanding" I missed before, and, at least for myself, I succeeded. And I hope that means other people will also be able to get a deep understanding of expected values as well.
NOTE: This StatQuest assumes that you are already familiar with the concept of an Expected Value. If not, check out the 'Quest: • Expected Values, Main ...
Also, if you want to learn more about Integration By Parts, here's a great resource: www.mathsisfun.com/calculus/i...
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0:00 Awesome song and introduction
1:30 Discrete and continuous variables defined
3:42 The exponential distribution
7:03 Approximating expected values with discrete intervals
10:25 Deriving the expected value formula for continuous variables
12:18 Calculating the expected value of an exponential distribution
18:17 Summary
#StatQuest

Пікірлер: 175

@statquest 2 жыл бұрын

Support StatQuest by buying my book The StatQuest Illustrated Guide to Machine Learning or a Study Guide or Merch!!! statquest.org/statquest-store/

@theburtmacklin9615 3 жыл бұрын

Amazing videos; if you only knew the way in which your channel had changed things for me… what started as a “hey this video looks cool” has largely been responsible for my wife enrolling into a data science postgrad program in the University of Waterloo.

@statquest 3 жыл бұрын

That's amazing! Wish your wife good luck for me and I hope it goes well. It's very exciting! BAM! :)

@Thedavidk 22 күн бұрын

@@statquest I did but the integral stuff was a bridge to far for me 🙂

@raulsena3917 2 жыл бұрын

Amazing videos, just wanted to thank you for all you have taught me. Now I am doing a Master's Degree in Artificial Intelligence and planned to do a Ph.D. and all thanks to your videos.

@statquest 2 жыл бұрын

Wow! That is awesome! Good luck!

@bogdankhamelyuk3857 Жыл бұрын

same bro. Also doing MS in AI and reminiscing some ML basics XD

@otavioaugustodeassuncao9858 Жыл бұрын

Currently i'm studying Statistics at Federal University of São Carlos (UFScar) and just wanted to thank you for all the helpful and fun content that you've been posting... Not only had helped me to understand but also has made me like Statistics even more

@statquest Жыл бұрын

Thank you very much! :)

@MrDanituga 3 жыл бұрын

One of the best explanations I have ever had! Congratulations on the video and on your content in general. Watch every video :D

@statquest 3 жыл бұрын

Thanks! 😃

@iroikoarakide8990 5 ай бұрын

Why do I have to attend college classes when you explain these concepts so thoroughly and in a third of the time? Great video as always man you are a Godsend

@statquest 5 ай бұрын

Wow, thanks!

@ahmadatta66 2 жыл бұрын

I'm in the library and I did not realize my laptop speaker was on, and the video started, as always, with a song. :D I'm not even embarrassed. Everybody should hear your songs, they are amazing haha

@statquest 2 жыл бұрын

BAM! :)

@josherickson5446 3 жыл бұрын

Triple bam bro! Now just need to refresh some calculus :)

@statquest 3 жыл бұрын

Yep!

@norukamo 4 ай бұрын

Just discovered this channel an hour ago and now I'm binging your videos. They're soo good and easy to understand. BAM!

@statquest 4 ай бұрын

Thank you very much! :)

@awaisjabbar8304 27 күн бұрын

Liked the video, before it even starts, awesome channel, why I have not heard of you before.

@statquest 26 күн бұрын

Thank you!

@newsupdates3622 3 жыл бұрын

Double bam! Here to watch another great statquest video for a refresher

@statquest 3 жыл бұрын

Thanks!

@zalavadiaridham 2 жыл бұрын

Amazing video josh!!! I'm waiting for the video on why we divide by n-1 when we compute the sample variance. Thank you for the very informative content that you put out.

@statquest 2 жыл бұрын

Thank you!

@hrig Жыл бұрын

yes! n-1please!!!!!!!!

@jaspermuller9851 5 ай бұрын

Is there any update on the n-1?

@dicksang2 3 ай бұрын

I am following through and want your version of n-1 as denominator to estimate the variance by sample variance. Thanks!!

@eduquest3169 2 жыл бұрын

Thanks a lot dear, may god bless you Josh.

@statquest 2 жыл бұрын

Thanks!

@durgeshkshirsagar116 3 жыл бұрын

You sound like kids' rhymes but when you teach, teach like a pro.

@statquest 3 жыл бұрын

Thanks! :)

@durgeshkshirsagar116 3 жыл бұрын

@@statquest welcome

@fedos 3 жыл бұрын

I think I've had too much to drink, because I read the notification as "Star Quest with Josh Starmer", and I was confused about your sudden shift to astronomy. I'm going to have to watch this tomorrow.

@Erosis 3 жыл бұрын

Star Quest would make a great April Fools joke. Have any astronomy friends, Josh?

@statquest 3 жыл бұрын

Once StatQuest makes me a billionaire, I'll make spaceship called Star Quest!!! You guys can come along for the ride.

@jerrycheung8158 2 жыл бұрын

Great video! May I ask where is the next video to go to for solving the n - 1 puzzle?

@statquest 2 жыл бұрын

Unfortunately I haven't made it yet. However, this is my favorite webpage on the topic: online.stat.psu.edu/stat415/lesson/1/1.3

@selva279 3 жыл бұрын

Eagerly waiting for your book...

@statquest 3 жыл бұрын

Working on it! :)

@sparshasherke250 Жыл бұрын

this explanation is magical :cries:

@statquest Жыл бұрын

@mohammedk.h.f3016 3 жыл бұрын

Thank you very much

@statquest 3 жыл бұрын

bam! :)

@kakashisensei8146 3 жыл бұрын

Josh, I'm really thankful to all your vdos, it's enlightening, lol. Please can you make some vdos on distribution functions (e.g., normal distribution)?

@statquest 3 жыл бұрын

I've got a basic video on the normal distribution here: kzfaq.info/get/bejne/qOB2i5h_vNXOoGQ.html and a video on maximum likelihood estimation with the normal distribution here: kzfaq.info/get/bejne/etRmlZyXqK-5oIE.html All of my videos can be found here: statquest.org/video-index/

@kakashisensei8146 3 жыл бұрын

@@statquest you're the best.. Thanks a lot!

@keshabdoley1727 3 жыл бұрын

You are an awesome singer As well as A great Teacher.🙃🙃🙃

@statquest 3 жыл бұрын

Thanks!

@Altamira_ Жыл бұрын

Thanks for all of your videos they are great. I'm waching the full playlist "statistics fundamentals" and have to admit that since I understood all of the previous videos thanks to your great explanations I was lost on this one when you used integrals :/ I think it's the video that requires the more calculcus knowledge.

@statquest Жыл бұрын

As long as you understand the discrete case, you should be good to go.

@Altamira_ Жыл бұрын

@@statquest Thanks :)

@alexandersmith6140 8 ай бұрын

Hi Josh. What video(s) after this should I watch to get closer to understanding why we divide by (n - 1) when finding the sample variance and sample covariance? An intuitive explanation for this seems nowhere to be found across the entirety of the internet, and the StatQuest channel has thus far been a divine gift of comprehension.

@statquest 8 ай бұрын

Have you seen this one: kzfaq.info/get/bejne/qa6Cdcpnp86vmn0.html Other than that, the best I can do is refer you here: online.stat.psu.edu/stat415/lesson/1/1.3 One day I turn that page into StatQuest, but not for a while.

@atg6174 5 ай бұрын

@@statquest why not now? i think its time to do so ...😢

@quanghoang3801 2 ай бұрын

@@statquest i have waited for 4 years 😢

@mamadi-uk7kv 2 жыл бұрын

Perfect man,Great

@statquest 2 жыл бұрын

Thanks a lot!

@mandaltech6249 2 жыл бұрын

Waiting for your book. You area real teacher.

@statquest 2 жыл бұрын

Thanks! I hope it is out in May!

@davonraymond3274 Жыл бұрын

appreciate you

@statquest Жыл бұрын

Thank you!

@user-lc6jq1hi1r 2 жыл бұрын

I am a simple man, I see integrals, I click. and thank God.

@statquest 2 жыл бұрын

bam!

@AG-cx1ug 3 ай бұрын

Did we finally get the final equation as at 7:40? Both are just area under the curve? Width wasn't being used before 11:00 - we were just using the specific outcome x, how did both width and specific outcome come in the final equation?

@statquest 3 ай бұрын

The equation at 7:40 is just the equation for the area under a curve defined by an exponential distribution, not the equation for the expected value of an exponential distribution. The equation for the expected value of an exponential distribution doesn't show up until 12:39. The big difference is that we multiply the formula for the exponential distribution by 'x', a specific outcome. Width isn't part of this equation because we take the limit earlier as the number of rectangles goes to infinity and their widths go to 0.

@homunculide8567 Жыл бұрын

Thanks for that and another videos with all that useful explanations... I'll keep binge watching. Regarding your explanation at 10:06... The second term and all following appear wrong to me, because the intervals are 10s each and not 20s, 30s, 40s... this the area of the rectangle should be likelihood x 10(s) for all terms (discrete rectangle) You are the expert but I can't wrap my head around that.

@statquest Жыл бұрын

I'm not sure I fully understand your equation. The area of each rectangle (the probability) is 10 x likelihood (obviously the values are approximated and rounded in the figure). The time, 's', should only be used to determine the likelihood at a specific point.

@sharan9993 3 жыл бұрын

When can i know why we divide sd by n-1? I hav seen ur videos on sd and know we need to subtract something but why 1? U said we need to know about expected values first. So when can i learn? Love ur videos. Thanks a lot

@statquest 3 жыл бұрын

Believe it or not, we still have a long way to go. Sorry this process is so painful!

@statquest 3 жыл бұрын

That said, if you want to skip to the head of the class, see: online.stat.psu.edu/stat415/lesson/1/1.3

@sharan9993 3 жыл бұрын

@@statquest ohhh i will wait bcoz i am curious yet lazy😂

@sharan9993 3 жыл бұрын

@@statquest i checked it out and it went over my head. Could u tell me the path that leads there? I will search and read them. Thanks a lot

@statquest 3 жыл бұрын

@@sharan9993 If I could, you wouldn't be waiting for my videos.

@pigritor 3 жыл бұрын

Dear Sir, could you do video on Linear Mixed models and GEE? Maybe you could create a separate donation target with threshold which you need to achieve to make one? It would be extremely useful. It is long waited and hard topic with a lot of contradictory info.

@statquest 3 жыл бұрын

I'll keep that in mind.

@maxyen9892 2 жыл бұрын

Could you comment on how we would find expected values for a normal probability distribution, given a mean and standard deviation?

@statquest 2 жыл бұрын

The expected value for a normal distribution is its mean.

@samieweee7468 5 ай бұрын

I am a bit confused is the value in y axis the likelihood of meeting a person after xth seconds or the number of people we meet after x th seconds cause during initial explanation the dots at each x were number of people we met after xth time.

@statquest 5 ай бұрын

What time point, minutes and seconds, are you asking about?

@samieweee7468 5 ай бұрын

@@statquest 2:37

@statquest 5 ай бұрын

@@samieweee7468 At 2:37, each dot represents an individual person. In other words, StatSquatch is creating a histogram. However, histograms have problems - gaps in the data and the data are not continuous (they have to be put in bins). Thus, we use an exponential distribution to approximate the histogram. The exponential distribution doesn't have gaps and is continuous. And, from there on out, we use the distribution, which gives us likelihoods on the y-axis, rather than the histogram, which gives us the number of people on the y-axis.

@user-ff5bi8ey1b 2 жыл бұрын

1) Why cant we just take average of all waiting times to get how much on average we would wait? 2) How often in real life we can use some sort of formula like exponential to describe a distribution? I think in most of the cases we can use NN to fit the distribution

@statquest 2 жыл бұрын

Expected values are helpful for doing Statistics and getting a sense of how likely future events will be. This is why we go through the trouble to do all this math instead of just taking an average or fitting an NN to the data.

@AG-cx1ug 3 ай бұрын

For each rectangle shouldn't we be multiplying the likelihood value by 10 and not 20,30? Since the area is just between 10-20 or 20-30?

@statquest 3 ай бұрын

At 9:00 we calculate the y-axis values (the likelihoods) for each rectangle by 10 because it is the distance between each tick-mark on the x-axis. This gives us the probabilities of each event. However, the events, or outcomes, themselves, are the x-axis values. So, starting at 9:10, we multiply the probabilities by the outcomes (the x-axis coordinates). When we add these products up, we get a weighted average of the outcomes (weighted by their probability of occurring). Is that what you are asking about?

@isurumahakumara 2 жыл бұрын

I couldn't find any quests on Time Series, If there aren't any would love to see one in the future!!!

@statquest 2 жыл бұрын

Me too! BAM! :)

@amirhosseinshafieian3951 2 жыл бұрын

THANK YOU for your great videos I just have a question, I think( I am not sure) that this data in this example follows Poisson distribution, not exponential!!!!! , am I right???

@amirhosseinshafieian3951 2 жыл бұрын

I think I got it my self

@statquest 2 жыл бұрын

Poisson is discrete and models something very differently from what we are modeling here with an exponential distribution. For details, see: en.wikipedia.org/wiki/Poisson_distribution

@adampotter760 10 ай бұрын

What the next video to watch to find out the elusive reason for dividing by n-1!? Please reply!

@statquest 10 ай бұрын

Unfortunately I don't have the video yet. In the mean time, check out: online.stat.psu.edu/stat415/lesson/1/1.3

@b1ueberrycheesecake Жыл бұрын

BAM!!

@statquest Жыл бұрын

@user-yp1rg2jr5z 2 жыл бұрын

Probability =( height*weight). Of rectangle How did you find that? Can you please explain how it indicates the probability? .. and also explain me that what is the value of "x" in continous random variables "mean"?

@statquest 2 жыл бұрын

The area under the curve between two points represents the probability of something happening between those two points (see: 4:48). This is simply how probability distributions are defined. We can solve for that probability exactly using calculus (see: 4:49), or we can approximate it using rectangles (height * weight).

@auzaluis 3 жыл бұрын

BAM!

@statquest 3 жыл бұрын

@aysan7513 2 жыл бұрын

omg you saved my life

@statquest 2 жыл бұрын

bam!

@taotaotan5671 3 жыл бұрын

probability is fun : )

@statquest 3 жыл бұрын

BAM! :)

@taotaotan5671 3 жыл бұрын

@@statquest I guess the next video will talk the variance of a random variable :)

@AG-cx1ug 3 ай бұрын

At 11:13 why is there still an x value at x * L(X=x) * width? Wasn't the x the width?

@statquest 3 ай бұрын

'x' is the value on the x-axis. It refers to a specific outcome or event happening at a specific time.

@SunSan1989 10 ай бұрын

I watched all the videos carefully, but why can't I find it? Why choose n-1 instead of N minus 2 or n-0.5?

@statquest 10 ай бұрын

Unfortunately I haven't made that one yet. In the mean time, check out: online.stat.psu.edu/stat415/lesson/1/1.3

@6872elpado 3 жыл бұрын

I didn't know you could call the integral of g'(x) the anti-derivative. Why didn't you just integrate g'(x) to get g(x)?

@statquest 3 жыл бұрын

"Antiderivative" and "Indefinite Integral" are synonyms. So we can say it either way. en.wikipedia.org/wiki/Antiderivative

@AG-cx1ug 3 ай бұрын

What if the data is erratic and doesn't fit with any of the lambda values to make the curve?

@statquest 3 ай бұрын

Then you can either use a different distribution, or you can try to approximate things with a histogram.

@AG-cx1ug 3 ай бұрын

@@statquest Thank you!

@tauqeeralitapya6447 Жыл бұрын

at 9:39, why are we multiplying 20 with 0.2 and not 10 as the interval is still 10 between 10 and 20?

@statquest Жыл бұрын

20 represents the specific, discretized, outcome. In other words, we have discrete time points, 10 seconds, 20 seconds, 30 seconds, 40 seocnds, etc. that we have to wait before we meet someone in StatLand. Since we have a histogram, everyone we meet between 0 and 10 seconds is categorized as "someone we met in 10 seconds". Likewise, everyone we meet between 10 and 20 seconds is categorized as "someone we met in 20 seconds". etc.

@priyankjain9970 11 ай бұрын

@@statquest then it should be like = (10*0.4) + (20*(0.4+0.2)) + (30*(0.4+0.2+0.1)) and so on.. and that's because for x=10 , probability will be area under curve till 10 which is 0.4, and for x=20 probability will be area under curve till 20 which is 0.63(roughly equal to 0.6) ... Please correct me if I am wrong

@ANGGAKAHFI-kr4yy Ай бұрын

Sorry sir i dont get it when u use 0.05 for the lambda,

@statquest Ай бұрын

What time point, minutes and seconds, are you asking about?

@ahmedabuali6768 3 жыл бұрын

Dear Professor, I would like to ask you if you have a very good lecture note for this book, introduction to mathematical statistics Robert v. hogg, I am ready to pay for that, I like your methodology in presenting.

@statquest 3 жыл бұрын

Unfortunately I've never read that book.

@ahmedabuali6768 3 жыл бұрын

@@statquest Ok, I would like to take this chance again to ask you if possible to add topics about Bayesian statistics. Many thanks and I am still following you :)

@statquest 3 жыл бұрын

@@ahmedabuali6768 I'm working on those videos right now. They should be out soon.

@ahmedabuali6768 3 жыл бұрын

@@statquest wow, very good, attached also the pdf presentation please, I am ready to buy it. Go ahead, dear Prof.:)

@alikhalili9961 2 жыл бұрын

Where is the n-1??????????? I am outraged. Lol. By the way, great video. Looking forward for more.

@statquest 2 жыл бұрын

Unfortunately this is just the first of many steps. :(

@Thedavidk Ай бұрын

Wow, that got complicated really quickly

@statquest Ай бұрын

Did you first watch the expected values for discrete variables: kzfaq.info/get/bejne/gbKjkpqTnLmvmWg.html

@megasun 8 ай бұрын

Number is not right around 8:25, the rectangle area should be approximate 0.3894. I was very surprised that the given rectangle area of 0.4 is bigger than the given integrated result of 0.39, because the rectangle area looks slightly smaller than the area under the curve ... still I love these videos. And of course, for the purpose of this lecture, having more digits here is distracting and not helpful.

@statquest 8 ай бұрын

Sorry if my inconsistent rounding through you off.

@oraciopozzo8694 2 жыл бұрын

When you approximate the expected value it is confusing that you add areas whose base is not the constant 10 interval. The areas of the rectangles should be in your example variable height given by the formula times 10 and not 10, 20, … Otherwise the rectangles are meaningless.

@statquest 2 жыл бұрын

I'm sorry that was confusing to you. However, each rectangle has the same width (as you can see in the illustration). However, each rectangle represents a different specific outcome. So 10, and 20 are not different widths, but different outcomes. 10 is the outcome represented by the first rectangle (with width = 10) and 20 is the outcome represented by the second rectangle (also with width = 10).

@priyankjain9970 11 ай бұрын

@statquest 11 ай бұрын

@@priyankjain9970 No, it's (10*0.4) + (10 * 0.2) + (10 * 0.1) + (10 * 0.09) + ... etc. To approximate the area under the curve, we add up the area of each rectangle. The area of each rectangle is the width (10) times the height (0.4 for the first, 0.2 for the second, 0.1 for the third, etc.).

@priyankjain9970 11 ай бұрын

@@statquest Thanks for reply. Actually the height is 0.04 for first, 0.02 for second, 0.01 for third and so on ( as explained by you @8.29 in video). Therefore area of rectangle will be 0.4 for first, 0.2 for second, 0.1 for third and so on. My concern is following As you stated E(X) = Σ x * p(X=x) .. This means E(X) = 10*(probability till 10) + 20 * (probability till 20) + 30 * (probability till 30) and so on. Now probability till 10 means area under curve till 10 which is = 0.4 probability till 20 means area under curve till 20 which is = 0.6 (approx) probability till 30 means area under curve till 30 which is = 0.7 (approx) Therefore E(X) should be (as per my understanding) = 10*0.4 + 20*0.6 + 30*0.7 + .... Please help me to understand this

@statquest 11 ай бұрын

@@priyankjain9970 Sorry about the typos with the area vs height. That said, the probability of observing an event between 10 and 20 seconds is not the cumulative probability of observing an event between 0 and 10 or between 10 and 20. Your equations use the cumulative probabilities, which is not correct in this situation. To clarify, the expected value is "the probability of observing an event between 0 and 10 seconds times the outcome, 10 (this is just the label for the any event that occurs between 0 and 10 seconds) + the probability of observing an event between 10 and 20 times the outcome, 20 (again this is just the label for any event that occurs between 10 and 20) + the probability of observing an event between 20 and 30 seconds time the outcome 30 + ....

@tedchirvasiu 2 жыл бұрын

5 years later: if you watched this video hoping to learn exactly why we divide by n-1, you are one step closer to understanding this mystery, but not quite there yet.

@statquest 2 жыл бұрын

Yep. Sorry it is taking so long.

@AG-cx1ug 5 ай бұрын

At 14:30 isn't the anti derivative just the integral?

@statquest 5 ай бұрын

yep

@pibruks 2 жыл бұрын

$$ #BAM!!!

@statquest 2 жыл бұрын

@satyamraj2779 4 ай бұрын

you are too cute with your bams!

@statquest 4 ай бұрын

@MrAzrai99 2 жыл бұрын

Just realizing the significance of the calculus I've learnt long time ago ":"(

@statquest 2 жыл бұрын

bam!

@MrAzrai99 2 жыл бұрын

@@statquest Please make a video on multivariate normal distribution next🙇‍♂️

@statquest 2 жыл бұрын

@@MrAzrai99 I'll keep that in mind.

@RoyalYoutube_PRO Ай бұрын

Has the mystery of 'n-1' been resolved yet?

@statquest Ай бұрын

Not yet. The best I can do is give you a link: online.stat.psu.edu/stat415/lesson/1/1.3

@tim40gabby25 3 жыл бұрын

09:50 Eyeballing, 0.4 no way is twice the height of 0.2

@statquest 3 жыл бұрын

Rounding errors.

@tim40gabby25 3 жыл бұрын

@@statquest Appreciate the response, tho you went on a bit. Old UK duffer here, enjoying the ride :)

@VladLanz 20 күн бұрын

Looks like the math is wrong for the approximation of expected value of distribution: 1) for each bin we compute: f(x) = lambda * e ^ (-lambda * x), where f(x) is the value of probability density function (PDF) for the middle point 2) we compute probability of each bin: p(x) = f(x) * delta, where delta is the width of the bin, in our case = 10 3) This is a step with mistake: we calculate contribution for each bin and sum everything up: E(x) = sum ( p(x) * x ), where x is the middle/average point for each bin, but in your video you took upper bound instead of an average value for each bin. If you do calculation this way, you get 19.23 which is closer to true value

@statquest 20 күн бұрын

I wouldn't say the math is wrong because the purpose is only to illustrate a concept, rather than how the math is actually done. In practice, we don't do a summation, we take the integral.

@VladLanz 20 күн бұрын

@@statquest sure, in practice we take the integral. But for approximation it makes more sense to take average value for each bin rather than it's top value. [5, 15, 25, ... 95] instead of [10, 20, 30, ... 100]. In case that's you and not some hired assistant who's answering comments here: thank you for your work, you're amazing🤗 You're the main reason i managed to remember everything i learned in university more than 10 years ago, started to master ml and deep learning and began working as data analyst

@statquest 20 күн бұрын

@@VladLanz That's me! Thanks! :) (and I still wouldn't say taking the edge is 'wrong' - different, and maybe it doesn't make as much sense for the sake of getting the best approximation, but not wrong).