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

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Unformatted text preview: 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 ) ≥ 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 that is, P ( a ≤ X ≤ b ) is the area under the curve...
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This note was uploaded on 09/07/2010 for the course FAS stat 230 taught by Professor Unknown during the Fall '08 term at American University of Beirut.

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

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