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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 Chapter4˙print.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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This note was uploaded on 03/30/2011 for the course STAT 511 taught by Professor Bud during the Spring '08 term at Purdue.
- Spring '08