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Unformatted text preview: Continuous Random Variables and Probability Distributions Chapter 4 2 Continuous Random Variables Continuous random variables can assume infinitely many values corresponding to points in a particular interval. A random variable X is continuous if its set of possible values is an entire interval of numbers (If A < B , then any number x between A and B is possible). If the measurement scale of X can be subdivided to any extent desired, then a variable is continuous, if it cannot, the variable is discrete. Examples of Continuous Random Variable If measurement scale can be subdivided to any extent possible, variable is continuous X = Depth measurement of a lake (A = minimum depth, B= max depth) X = pH of a chemical compound (range 0 14, but any value in between is possible) Discrete measurements often wellmodeled by continuous functions 0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015 0.02 0.025 0.03 Say you measured the depth of a lake to the nearest meter and then measured it to the nearest cm and built probability histograms: limit A B Meters Cm Limit of a sequence of discrete histograms (probability density function) 0.005 0.01 0.015 0.02 0.025 0.03 A B f(x)= 9x 2 6 Probability Distribution Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f ( x ) such that for any two numbers a and b , ( 29 ( ) b a P a X b f x dx ≤ ≤ = ∫ The graph of f is the density curve . 7 Probability Density Function For f ( x ) to be a pdf 1. f ( x ) > 0 for all values of x . 2.The area of the region between the graph of f and the x – axis is equal to 1. Area = 1 ( ) y f x = 8 Probability Density Function is given by the area of the shaded region. ( ) y f x = b a ( ) P a X b ≤ ≤ 9 Uniform Distribution A continuous rv X is said to have a uniform distribution on the interval [ A , B ] if the pdf of X is ( 29 1 ; , 0 otherwise A x B f x A B B A ≤ ≤ = A B 1 f(x) A B pdf of Uniform Distribution Area = 1 x Example… A read/write head for a hard disk has to locate a record as the disk rotates once every 25 ms. Let X = time it takes to locate the record. Assume that X is uniformly distributed on the interval [0,25]. What is: P(10 < X < 20)? P(X > 10)? 11 For a Nonuniform Distribution: Suppose the error involved in making a measurement is a continuous rv X with pdf: f(x)= .09375(4x 2 ) 2 < x < 2 0 otherwise Compute P(X> 0). Compute P(1< X< 1). Compute P(.5< X`OR X > .5). 12 13 Probability for a Continuous rv If X is a continuous rv, then for any number c , P ( x = c ) = 0. 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 = < < Example… 14 The Cumulative Distribution Function The cumulative distribution function, F ( x ) for a continuous rv X is defined for every number x by ( 29 ( ) ( ) x F x P X x f y dy∞ = ≤ = ∫ For each x , F ( x ) is the area under the density curve to the left of x . 15...
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This note was uploaded on 04/17/2011 for the course ENGR 0020 taught by Professor Rajgopal during the Spring '08 term at Pittsburgh.
 Spring '08
 Rajgopal

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