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Unformatted text preview: STAT511 Summer 2009 Lecture Notes 1 Chapter 4 June 28, 2009 Continuous Random Variables & Probability Distributions 4.1 Continuous Random Variables Chapter Overview Continuous random variables Probability density function (pdf) Definition and interpretation Cumulative distribution function (cdf) Definition and interpretation Relationship between cdf and pdf Expectation, variance and percentile for continuous rv Some continuous distributions Uniform and exponential Normal * Pdf * Using normal table Other continuous distributions- Gamma, Chi-squared, Weibull, Lognormal, Beta etc. Continuous rv and the Probability Density Function Continuous random variables Definitions Examples Probability density functions (pdf) Definitions Interpretations Examples Continuous rv Definition 1. A random variable X is said to be continuous if its set of possible value includes an entire interval of numbers on the real line. Example 4.1.1 (Example 4.1 in textbook) Make depth measurements at a ran- domly selected location in a specific lake. Let X = the depth at this location. X can be any value between 0 and maximum depth M . Example 4.1.2 (Example 4.2 in textbook) A chemical compound is randomly selected and let X = the pH value. X can be any value between 0 and 14. Purdue University Chapter4print.tex; Last Modified: June 28, 2009 (W. Sharabati) STAT511 Summer 2009 Lecture Notes 2 Probability Density Function (PDF) Definition 2. Let X be a continuous rv. Then a probability distribution or proba- bility density function (pdf) of X is a function f ( x ) such that for any two numbers a and b with a b , P ( a X b ) = Z b a f ( x ) dx. The graph of f is the density curve. i.e., the probability that X falls in [ a,b ] is the area under the function f ( x ) above this interval. f ( x ) must satisfies the following: 1. f ( x ) 0 for all x . 2. R - f ( x ) dx = 1 Probability Density Function (PDF) Interpretations of f ( x ) The density function f ( x ) gives us an idea about the distribution of probability density instead of probability itself. 1. For any c P ( X = c ) = 0, i.e., the probability that X takes any specific value is 0. 2. We can only look at the probability that X falls on a specific interval. This is given by the integration of f ( x ). 3. For any two numbers a and b with a < b , P ( a X b ) = P ( a < X b ) = P ( a X < b ) = P ( a < X < b ) = R b a f ( x ) dx . Pdf Examples Example 4.1.3 Uniform Bus comes every 20 minutes, let X = waiting time till a bus comes. The pdf of X is: f ( x ) = 1 20 , x 20 . What is the probability that waiting time is longer than 5 minutes? What is the probability that the waiting time is between 5 and 10 minutes? In general Given b > a , X with pdf: f ( x ) = 1 b- a ,a x b is said to have uniform distribution....
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- Spring '08