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Unformatted text preview: MA/STAT416: Probability Lecture Notes Spring 2011 1 Chapter 7: Continuous Random Variables April 1, 2011 7 Density Functions, CDF, Quantile Function 7.1 The Probability Density Function (pdf) and Cumulative Distribu- tion Function (CDF) Definition. Let X be a real-valued random variable taking values in R . A function f ( x ) is called a probability density function if and only if f ( x ) ≥ ∀ x ∈ R , and Z ∞-∞ f ( x ) dx = 1 . Note: P ( a ≤ X ≤ b ) = Z b a f ( x ) dx. Definition. Let X be a real-valued random variable with a pdf f ( x ). Then, the cumulative distribution function of X is defined by F ( x ) = P ( X ≤ x ) = P ( X < x ) = Z x-∞ f ( t ) dt. Remark. f ( x ) = F ( x ). Definition. Let F ( x ) be the CDF of X . The median of X is the number m such that F ( m ) = 0 . 5 . Definition. Let X have the CDF F ( x ). Let 0 < p < 1. The p th quantile of X is defined to be the first x such that F ( x ) ≥ p .: F- 1 ( p ) = infx : F ( x ) ≥ p....
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- Spring '08
- Probability, Probability theory, probability density function, Cumulative distribution function, CDF, quantile function