# What are Continuous Random Variables? → Probability & Statistics ## Метаданные - **Канал:** Socratica - **YouTube:** https://www.youtube.com/watch?v=ajrFIsv0bNI - **Дата:** 09.04.2026 - **Длительность:** 8:00 - **Просмотры:** 3,826 ## Описание 𝙎𝙄𝙂𝙉 𝙐𝙋 for tastefully infrequent updates about our upcoming Probability & Statistics course. 💌 https://snu.socratica.com/probability-statistics-course How do you calculate probabilities for a random variable with infinitely many outcomes? Your instinct may be to reuse what you learned when modeling coin flips and dice rolls. But transitioning from the finite to the infinite requires a shift in thinking. The math of discrete random variables and continuous random variables are fundamentally different. Today, let's explore the infinite world of continuous random variables. We'd like to send a special thank you to our VIP Patrons at Patreon! 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Ready to 🧠 𝙇𝙀𝘼𝙍𝙉 𝙈𝙊𝙍𝙀 with Socratica? 📺 𝙎𝙪𝙗𝙨𝙘𝙧𝙞𝙗𝙚 for SMART videos in Math, Science & Programming: http://bit.ly/SocraticaSubscribe ▶️ 𝙋𝙇𝘼𝙔𝙇𝙄𝙎𝙏𝙎: Study Tips http://bit.ly/StudyTipsPlaylist Python http://bit.ly/PythonSocratica Chemistry http://bit.ly/Chemistry_Playlist Calculus http://bit.ly/CalculusSocratica Geometry http://bit.ly/GeometrySocratica #probability #discretedistributions #randomvariables ## Содержание ### [0:00](https://www.youtube.com/watch?v=ajrFIsv0bNI) Segment 1 (00:00 - 05:00) What's the difference between rolling a marble and a standard six-sided die? When you roll the die, you're trying to predict which side will face up. There are six possible outcomes, and if the die is fair, each one has a one in six chance. With the marble, however, things are very different. There are infinitely many points on its surface that could end up on top. Because there are a finite number of outcomes with a six-sided die, that random behavior is called a discrete random variable. The marble, on the other hand, has a smoothly varying infinite set of outcomes. This is a continuous random variable. When you roll the marble, what's the probability that a specific point ends up on top? You might be tempted to say one out of infinity, which would be zero. But this doesn't pass the smell test, does it? When you roll the marble, one of the points does actually wind up on top. How can something with a 0% chance of happening actually happen? This is the central challenge with continuous random variables. Your intuition wants to reuse the tools from discrete probability, assigning a probability to each outcome, but that approach breaks down here. Part of the issue is infinity. In common usage, the world is split into the finite and infinite, but in mathematics, there is more than one kind of infinity. The two types of infinity you need to be familiar with are countably infinite and uncountably infinite. If you can list outcomes one by one, even infinitely many of them, they're countable. But if there are too many outcomes to list, even in principle, then they're uncountable. It may help to think of it this way. When outcomes can be listed and assigned individual probabilities, we're in the world of discrete random variables. When outcomes vary smoothly across a continuum, we need a different approach. That's the world of continuous random variables. If you want to improve your odds of learning probability, sign up for our course. Look for the link below. Examples of continuous random variables include the rolling marble, the high temperature tomorrow, and the maximum height you achieve in your life. All three have an uncountable number of outcomes. Do you believe that? For the temperature, you may be thinking it's always an integer. That's not true. If you use more and more precise scientific equipment, you will find you can add more and more decimals to the temperature for a more precise value. Similarly with your height. When measuring, we usually round our height to the nearest inch or centimeter, but if you were to use a sophisticated laser measuring device, you could make a much more precise measurement. With more precise instruments, you can measure more and more decimal places. We usually round up or down, but in reality, these random variables vary continuously. So, how do we measure probabilities for continuous random variables? What's the continuous version of a probability mass function? Let's return to our rolling marble. Because there are an uncountably infinite number of points on the surface of this sphere, it doesn't make sense to assign a probability to each point, to each outcome. If each point had any positive probability, the total would blow up past 100%. And if each point has probability zero, adding them up still doesn't get us to 100%. Either way, the approach fails. The trick is to assign a probability to regions, not points. For example, imagine if we painted the top half of our marble red and the bottom half blue. Then when you roll the marble, it stands to reason there's a 50% chance a red point will end up on top and a 50% chance of rolling a blue. Or perhaps you could paint the marble so that the top third is red, the middle third is green, and the bottom third is blue. Then when you roll the marble, there's a 33% chance of each outcome. We're rounding slightly. We can generalize this idea. Suppose you draw a squiggly region on the marble and paint it white, leaving the rest black. What's the probability that a white point ends up on top? First, we need two things: the area of the squiggly region and the surface area of the sphere. The formula for a sphere is 4 pi r squared, where r is the radius of the sphere. Let's assume the radius is 1 cm, so the surface area is 4 pi cm squared, which is about 12. 57 cm squared. Suppose the area of the white region is 3 cm squared, then the probability of rolling white is the fraction of the sphere that's white, which is 23. 87%. black is 76. 13%. We could write the general formula like this, for a marble of radius r and a region of area a. Then the probability of a point inside the region being on top is this: the fraction of the surface area that's white. Another way you can ### [5:00](https://www.youtube.com/watch?v=ajrFIsv0bNI&t=300s) Segment 2 (05:00 - 08:00) say this is the probability is the area of the region divided by total surface area. The rolling marble example allowed us to focus on one important part of probabilities of continuous random variables. You measure the probability of regions, not specific outcomes, but with the marble, we failed to address another common feature of continuous random variables: bias. With a perfectly smooth marble, no points were more or less likely than others to end up on top. Every point was equally likely because the marble was perfectly uniform. However, with most continuous random variables in the real world, some outcomes are more likely than others. Consider the weather. Suppose today is July 1st and you live in Atlanta. It was 85° today and the forecast for tomorrow is 87. Which is more likely, that tomorrow is in the 90s or that it's snowing in the 20s? Clearly, hot weather is far more likely than snow in summer. So, how do we express that mathematically? We use a function called a probability density function. This is a function f that assigns a density, not a probability, to each possible value. If you graph this function, you might get a familiar shape, the bell curve. The horizontal axis represents temperature, and the vertical axis is the density. The peak occurs around 87, the forecast. It has the highest density, which means it's the most likely. That also means temperatures near 87 are more likely than temperatures far away. But remember, there are an uncountable number of possible temperatures tomorrow, so we cannot assign a probability to it being 87°. The probability of exactly 87° is still zero. However, we can find the probability of regions and ranges of outcomes. The probability that it's in the 90s is the area under this curve from 90 up to 100°. This is the big idea for continuous random variables. A density function allows you to mathematically say that some outcomes are more likely than others without assigning a specific probability to each individual outcome. The function assigns a density to each possible outcome, not a probability. To find the probability of a range of outcomes, you find the area under the density function over this region. If you want to improve your odds of learning probability, sign up for our course. Look for the link below. --- *Источник: https://ekstraktznaniy.ru/video/52898*