Chapter 4 2114

Chapter 4 2114 - Continuous Random Variables and...

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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 well-modeled 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)= 9-x 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 Non-uniform Distribution: Suppose the error involved in making a measurement is a continuous rv X with pdf: f(x)= .09375(4-x 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.

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Chapter 4 2114 - Continuous Random Variables and...

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