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chapter3 - STAT 230 Introduction to Probability and Random...

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STAT 230: Introduction to Probability and Random Variables Summer 2009 Chapter 3: Random Variables of the Continuous Type 1 PDF and CDF Thus far, we have only discussed discrete random variables in detail. Recall that discrete random variables describe things that can be counted (number of successes in a sequence of Bernoulli trials, etc). There is another class of random variables that describe things that can be measured but not counted. Examples are height, weight, length, area, volume, etc. lifetimes, waiting times, etc. These continuous random variables can take values in an interval, unlike discrete random variables that take values in a countable set, like 1, 2, 3, . . .. Recall that with a discrete random variable X we had a PMF p X ( x ) 0 and all prob- abilities are found by summing this function over suitable values of x . Things are basically the same for a continuous random variable X : it has a function f X ( x ) 0, called the PDF and all probabilities are found by integrating over suitable values of x . Definition 1. Probability density function (PDF) A random variable X is called continuous if there is a function f X ( x ) , called the probability density function (PDF), that satisfies the following properties: (i)- f X ( x ) 0 for all x . (ii)- R -∞ f X ( x ) dx = 1 . (iii)- For any constants a < b , the probability that X is between a and b is: P ( a X b ) = R b a f X ( x ) dx 1
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that is, P ( a X b ) is the area under the curve f
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