Continuous probability distribution intro

Summary

In this video, Khan Academy explains the basics of continuous probability distribution through practical examples. A continuous random variable can take on any real value within a range, and its probability distribution can be represented by a probability density function (PDF). The video delves into calculating probabilities for different intervals, demonstrating how the probability of pinpointing an exact value (e.g., exactly three) is effectively zero due to its infinitesimal width. Such explorations highlight the conceptual difference between discrete and continuous distributions.

Highlights

• Continuous random variables have no discrete steps; they flow smoothly across the range. 🌊
• Calculating the probability for a range involves finding the area under the PDF curve for that interval. 📐
• As the interval around a specific value shrinks, the probability approaches zero. 🧐

Key Takeaways

• Continuous random variables can take on any value within a range, as opposed to discrete variables. 📊
• Probability distributions of continuous variables are represented using a probability density function (PDF). 📈
• The probability of a continuous variable being exactly a certain value is zero due to zero width. 0️⃣

Overview

Sal from Khan Academy provides a clear and engaging overview of continuous probability distributions, focusing on the concept of probability density functions (PDFs). The continuous random variable he uses is confined to taking values between 0 and 5, demonstrating how uniform probability density functions work in such scenarios.

By using simple numeric examples, he illustrates the fundamental way probabilities are calculated over intervals in a continuous context—it's all about the area under the curve! Sal emphasizes how shrinking these intervals to pinpoint an exact value brings the probability down to zero, which is a crucial insight into the nature of continuous distributions.

The video methodically walks through practical probability problems, showing viewers the mathematical logic behind these computations. It also underscores the fascinating realization that in a continuous setting, the probability of a specific value is nil, smoothly guiding viewers from basic concepts to this somewhat counter-intuitive conclusion.

Chapters

• 00:00 - 00:30: Introduction to Continuous Probability Distributions Chapter 1 - Introduction to Continuous Probability Distributions: This chapter introduces the concept of continuous random variables and explores their probability distributions. It explains the process of constructing a probability distribution for a continuous random variable, using a graph with horizontal axis representing the values X can take, and the vertical axis representing the probability density of each value.
• 00:30 - 01:30: Understanding Probability Density Function (PDF) This chapter introduces the concept of Probability Density Function (PDF) with a focus on understanding why it is termed as 'density'. It begins with establishing that the random variable X cannot take on negative values, thus the probability density for any negative value is zero. Then, it illustrates a uniform density for the values between 0 and 5. The chapter emphasizes understanding the density aspect of the function and invites further exploration into its properties.
• 01:30 - 02:30: Characteristics of Continuous Probability Distributions This chapter discusses the characteristics of continuous probability distributions. It emphasizes that continuous random variables can take any value within a specified range. Specifically, in this chapter's example, it mentions that there is zero probability for values greater than 5 or less than 0, with a uniform probability distribution between these bounds (0 and 5). Additionally, it describes how continuous distributions can be represented by a probability density function.
• 02:30 - 03:30: Calculating Probabilities for Ranges of X The chapter discusses the concept of a probability density function (PDF) with a focus on uniform PDF. It challenges the reader to consider the constant height or level of the uniform distribution and where this horizontal line intersects the vertical axis.
• 03:30 - 04:30: Exploring the Uniform Probability Density Function The chapter 'Exploring the Uniform Probability Density Function' deals with the concept of probability density functions (PDFs). It emphasizes the fundamental principle that for both continuous and discrete random variables, the sum of all possible outcomes must equal one, encapsulating a 100% certainty of obtaining one of the possible outcomes. For discrete random variables, this involves summing the probabilities of distinct outcomes, while for continuous random variables, it involves integrating over a range.
• 04:30 - 05:30: Philosophical Insights into Continuous Probability The chapter delves into the philosophical aspects of continuous probability, emphasizing that continuous random variables can assume any value within a given range, stretching infinitely, including irrational numbers like π (pi), e, and the square root of two. This characteristic allows for an infinite number of possible values that a random variable might take, illustrating the concept of probability density in continuous scenarios.
• 05:30 - 10:00: Conclusion: Probability of Specific Points in Continuous Distributions This chapter discusses the concept of probability in continuous distributions, emphasizing the importance of the area under the probability curve. The area represents the combined probability of all possible outcomes and must equal one for a probability density function to be valid. The chapter illustrates this with an example, considering a base length of a distribution and calculating the necessary height for the area to maintain the property of unity.

Continuous probability distribution intro Transcription

• 00:00 - 00:30 let's say I have some random variable X and it is a continuous it is a continuous random random variable and I want to explore its probability distribution in fact I want to construct a probability distribution for it so let's draw here on the vertical or on the horizontal axis I should say these are the values that X can take on and in the vertical axis I'll essentially say the probability density for each of
• 00:30 - 01:00 those values and we'll see we'll discover in a few moments why we are calling it density so let's say that my random variable X it can never take on negative values so it's a zero probability density of taking on any value that is negative and then it has a uniform density of taking on any value between 0 & 5 so let's say that that's 0 say this is 1 2 3 4 5 and then it can't
• 01:00 - 01:30 take on any value above 5 so it's not going to as a zero probability of any value any value greater than 5 0 probability of any value less than 0 and uniform probability between 0 & 5 so this is a shaded encircle uniform probability between 0 0 & 5 so this right over here when we're talking about a continuous probability distribution this can also be referred to as a probability density function probability
• 01:30 - 02:00 density function sometimes a PDF probability density function in this case you might notice it is a uniform it is a uniform probability density function now the first question I have for you is what is the height or what is this level we see it's uniform but uniform at what level what is this value going to be where this horizontal line intersects the vertical axis for our
• 02:00 - 02:30 probability density function well to think about it we just have to realize that whether we're talking about a continuous random variable or a discrete random variable the sum the probability that you get any one of the possible outcomes the sum of all of those have to be equal to one you have a hundred percent chance of getting one of the possible outcomes for your random variable whether it's discrete or continuous so the sum in the case of a discrete we kind of summed up the bars in a continuous random variable we have to
• 02:30 - 03:00 realize it can take on any value not just one or two or three it could take on it could take on 3.14159 keeps going on and on and on forever it could take on the value of pi' it could take on it could take on the value of two point seven one gone and on and on the value the number e it could take on square root of two and any number in between so when we're thinking about all of the possible scenarios all of the possible values that our random variable can take on times the density the probability
• 03:00 - 03:30 actually is now the area the combined probability of all of the possibilities are now is now this area so for this random variable in order for this to be a legitimate probability density function or probability distribution the area here that I've highlighted in orange this area here needs to be equal to one so this area needs to be equal to one and given that this base here is of length five what does this height need to be well five times what is going to
• 03:30 - 04:00 be equal to one five times its reciprocal so we have a uniform density right here at 1/5 so given that we've defined this probability density function in this way let's think about some probabilities so what if I were to ask you what if I were to ask you the probability the probability that X is greater than one let's say greater than or equal to 1 and less than or equal to
• 04:00 - 04:30 2 what is this probability going to be equal to well you just have to say well what are all the possible values that X can take on so it can be between 1 and 2 including 1 and 2 and so here is its combined probability that X is in that range it's going to be the area under the curve under the curve under the under the curve in that range and so what is this area well the base here is one the height here is 1/5 we haven't drawn the height to scale here base here
• 04:30 - 05:00 is one height is 1/5 so it's going to be 1 times 1/5 which is equal to 1/5 let's think about another one let's think about another one what is the probability that our random variable is greater than or equal to 4 and less Center equal to 4 and 1/3 what's that probability going to be equal to so once again what's the range we can be greater than or equal to 4 greater than equal to
• 05:00 - 05:30 4 and lepton less than or equal to 4 and 1/3 which is right about there so we really care about is the area under the curve under the curve in this range and lucky for us this is a rectangle so the base between 4 & 4 & 1/3 you have a distance of 1/3 and then the height once again is 1/5 times 1/5 is equal to 1 over 15 now let's do something interesting now let's do something interesting what is the probability and
• 05:30 - 06:00 not that the other stuff wasn't that interesting but let's do something even more interesting what is the probability what is the probability that X X is greater than or equal to 3.9 and less than or equal to actually let me do it this way 2.9 let's say X is greater than or equal to 2.9 or 2 point 9 is less than or equal to our random variable which is less than or equal to 3 point 1 that didn't look like an X and is less than or equal to 3 point 1 what is this
• 06:00 - 06:30 probability going to be equal to so we have this little range here right over here the height is 1/5 we've seen that over and over again but what's the area of this rectangle well the base here is between 2 point 9 and 3 point 1 so that is point zero 2 let me draw that rectangle a little wider so if we dot like this draw it zoom in a little bit this point right over here is 2.9 at this point right over here is 3 point 1 the difference between the two is 0.2 or
• 06:30 - 07:00 you could say it's 1/5 so 0.2 and then the height here is 1/5 so the base is 0.2 or 1/5 and then the height is 1/5 so it's 1/5 times 1/5 is equal to 1 over 25 well you say well how is that any more interesting what we just did well let's escalate a little bit what's the probability not of that range let's take the probability let's take the probability that 2.99
• 07:00 - 07:30 is less than our random variable which is less than the less than or equal to two point nine nine is less than or equal to our random variable which is less than or equal to three point zero zero one what is this going to be equal to so now we've made our range a little bit smaller our base now is now going to be point zero two the difference between three point zero one and two point nine nine is point zero two so it's now point zero to base and the same height 1/5 so
• 07:30 - 08:00 the base is now one not one-fifth but one 50th that's the same thing as point zero two and we multiply that times 1/5 times 1/5 gives us one let me scroll over to the right a little bit 1 over 250 and we could keep going we could keep going and I think you see why this is getting interesting what's the probability that two point nine nine nine is less than or equal to our random variable which is less than or equal to three point zero zero one what's this
• 08:00 - 08:30 going to be equal to well same exact logic the range the base this will range right of our random variable it is now a range of one five hundred point zero zero two one five hundred so it's now going to be one over five hundred one over five hundred times the height times 1/5 1/5 which now gives us one 2,500 one 2,500 so you see we're getting closer and closer to X being exactly three and
• 08:30 - 09:00 our probabilities getting lower and lower and lower as we narrow our range we can start getting really really really close to 3 so with that let's just finish this video with a very philosophically interesting question what is the probability that my continuous random variable defined this way what is the probability that it is a exactly exactly equal to three not three point zero one not two point nine nine nine nine nine nine nine nine what is the probability that is exactly equal to
• 09:00 - 09:30 three well now our rectangle has essentially degenerated or it's just degraded down to just a vertical line its height is still one fifth but it has absolutely no width it has absolutely no width it is a R it is an infinitely skinny rectangle and so your probability here it has no area it has no area so the probability is zero so you actually have a zero probability of getting exactly three not two point nine nine
• 09:30 - 10:00 nine not not between two point nine nine nine nine nine nine nine and three point zero zero zero one we're talk about infinite precision getting exactly three the probability is zero and hopefully this little progression that we saw gives you an indication of why that is is we get to a tighter and tighter range around three the probability was getting closer and closer to zero