Overview
On this page, we start investigating how to model probabilities for a continuous random variable, using the cumulative distribution function (cdf) and probability density function (pdf).
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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Use the cumulative distribution function of a continuous random variable to calculate the probability it takes a value in an interval \(I \subseteq \mathbb{R}\).
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Use the probability density function of a continuous random variable to calculate the probability it takes a value in an interval \(I \subseteq \mathbb{R}\).
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Understand the relationship between the cumulative distribution function and the probability density function of a continuous random variable, and be able to find each from the other.
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Use the method of distribution functions and the method of substitution to derive the probability density function of a random variable \(Y = g(X)\) where \(X\) is a continuous random variable with a known probability density function and \(g: \mathbb{R} \to \mathbb{R}\).
To prepare for class
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Watch the following video (by 3Blue1Brown) which gives an introduction to the idea of a probability density function for continuous random variables:
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Watch the following video (by Brendan Cordy) which explains how to use the cumulative distribution function to get a good model for the probability distribution of a continuous random variable, and how to obtain the probability density function from this:
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Watch the following video (by Brendan Cordy) which shows an example of how to work with the probability density function of a continous random variable:
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Watch the following video (by jbstatistics) which shows some example calculations involving the pdf and cdf of a continuous random variable (including a calculation of the so-called median):
After class
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Watch the following videos (by Brendan Cordy) which explain two methods of evaluating probabilities for functions of a continuous random variable, the method of distribution functions and the method of substitution:
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Review general facts about integration as needed using these Calculus 2 review pages:
