An Introduction to Continuous Probability Distributions

Let’s look at an introduction to
continuous random variables and continuous probability distributions. Continuous random variables can take on an
infinite number of possible values, corresponding to every value in an interval. So for example we might have a random variable
that takes on any value between 3 and 4. Our random variable might take on the value
3.1, or 3.12, or 3.1278694, or 3.8, or what have you. Any of the infinite number of values in between 3 and 4. Or a common one that we see, we have a random variable taking on any
positive value, so any value between 0 and infinity Here is approximately the distribution
of the height of adult Canadian males. Now height is a continuous random variable and it’s going to have a continuous probability distribution. And this looks something like a smooth
version of a histogram. A little loosely speaking, values of the
variable where the curve is high are more likely to occur than where it is low. So here we would be more likely to get a height in this range than way out here in this extreme. We cannot model continuous random
variables with the same methods we used for discrete random variables. There will be some similarities, but we
will have to use different methods. We model a continuous random variable
with a curve, f(x), called a probability density function or pdf. Here’s another example of a
continuous probability distribution, the distribution of time to failure in
thousands of hours, for a type of light bulb. Values the random variable can take on are given down here on the x axis, and
the probability density function f(x) is a function giving the height of the curve at those values of x. f(x) represents the height of the curve at point x. An important notion is that for continuous random variables, probabilities are areas under the curve. Here’s a continuous probability
distribution for a random variable X. And the height of the curve is
represented by f(x), and the probability the random variable X falls in between two values a and b, is simply the area under the curve
between a and b. One important notion here is that
the probability the random variable X is exactly equal to any one specific value is 0. We could say the probability the random variable X
is equal to the value a is 0 for any a. We could think of the point a here as an infinitesimally small point with infinitesimally small area above it we call that area 0, So the probability that X is equal to
a for any a is 0. So for any continuous probability
distribution, let’s say the probability that X is equal to 3.12, that’s going to be equal to 0. So from a practical point of view it’s
only going to make sense to talk about the random variable X falling in an interval values. One implication of what we just talked about here is that this probability would be the
same as saying the probability that the random variable X is greater
than or equal to a and less than or equal to b. We can switch less than or equal to with
less than, it doesn’t matter because the
probability the random variable X is exactly equal to one specific value is 0. For any continuous probability distribution,
f(x) has to be at least 0 everywhere. Note that there is no upper bound on it, it can take on values greater than 1. One restriction is that the area under
the entire curve is equal to 1. And so these two restrictions ensure
that all probabilities lie between 0 and 1, and the probability of something happening is 1. There are a number of common
continuous probability distributions that come up frequently in theory and practice. One very common and extremely important
continuous probability distribution is the normal distribution. And it looks like this. This is the continuous uniform distribution for which f(x) is constant over the range
of possible values of x. The exponential distribution
look something like this. This is something we might see in
exponential decay or a number of other spots. And there are many other continuous probability distributions that are very important to us in probability and statistics. Probabilities and percentiles are found by
integrating the probability density function. Probabilities are areas under the curve, and areas under the curve are found
using integration. Fortunately for us statistical software
will carry out the integration for us in a lot of situations. And so in practice will be using statistical software or statistical tables to find these areas. Deriving the mean and variance of the probability
distribution also requires integration. Depending on your needs you may or may not need to know how to actually carry out the integration, so I’m going to look at those concepts in
separate videos.

Stephen Childs


  1. Thank you very much for your videos! I will definetly spread your channel in my university 🙂

  2. I really do not have enough words to thank you. I just want to tell you that I got a high mark in introduction to probability and I owe you in lot for that :), of course I will study in next semester "Introduction to inference" and "Introduction to the sampling" and i will be waiting for your videos ..Thank you very much =)
    your student from Oman

  3. You are very welcome! I'm glad to hear you got a good mark in your introduction to probability course! I have many videos on various topics in inference (and I'll be adding more) so you may very well find some that help you in your future courses. I'm very happy to do my little bit to help a few people around the world. Best wishes from Canada!

  4. The video is great! but I still have something that is not clear for me you might can help me. Let's say that I have a continuos cumulative F(X), suppose F(0)=O, F(0)>0 and F(0)<0 what is that meaning in terms of cumulative distribution and what's the relationship with the density function in that cases! 

  5. Thank you for the whole videos!  Even if i'm french speaking and so i don't perfectly understand all, your videos stay much better than the others !

  6. Your videos were a total lifesaver. Thank you. Excellent teaching abilities 

  7. I'm a nurse and in a beginning statistics class. It's killing me. We move very fast and understanding how to use all the formulas is tough for me. This video is helpful. I will try to track with you as I go through upcoming subjects.

  8. @jbstatistics If area under the curve represents – PROBABILITY, what does the y-value represent, It is very confusing please reply

  9. im at a lost for words,when i think abt how much i appreciate ppl like u..thnks for making these videos.Your helping ppl acheive their goals…regardless of hw small a topic like this is. Thanks and Bless you. #from Jamaicawithlove

  10. I don't completely understand why the area under a graph represents probability of something happening. Extending from discrete variables, their probability was not the area under the graph but rather just the corresponding value on the y-axis, why should it be any different here? I can intuitively see why any 1 value would have the probability of 0, but even then, where does the area under the graph come in?

  11. How many attempts does it take to complete one of your videos? Do you record all in one go? Nicely done!

  12. I've learned more from your videos in a few hours than I've learned from my professor in weeks…can't state enough how well you explain things in an easy to understand way, without diminishing the knowledge that is still expressed.

  13. You are a super saviour.. You save degrees, you build confidence in thousands, to know and grow. Tht's what is a teacher… I literally stay on leaves to finish your courses comfortably. Hats off!!

  14. I study aerospace engineering at a pretty damn good university and you explain this material better than any professors or TA's around. Dank je wel 🙂

  15. Great video. But what I don't get is why the area under the curve stands for the probability between two elements of the random variable

  16. Thank you so much, you helped me pass probability, those proffesors only come with shit, you made it all clear to me!

  17. Now with just real world observations, how do you actually know f(x) to even draw the curve first. Only once you know f(x) you can get area under the curve. Now the real world observations may or may not fall into a normal distribution.

  18. This man is an Angel. Really need these type of people in my life. You make statistics look like a piece of cake for me.

  19. NORMAL DISTRIBUTION CURVE ranges from 0 to infinity .?? or… + ,- infinity. anyone?

  20. Can a probability density function be discontinuous? Like, could the chances of a random variable assuming a certain value be greater or smaller by a significant amount than the values immediately next to it?

  21. I have a final in two days that I would die to if I didn't have these videos. God bless

  22. This is hands down the best explanation I've come across youtube on this topic!

  23. Just came back to say that I passed my statistics/probability exam because of you. Thank you!

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