Probability distribution, continuous random variables

Probability distribution, continuous random variables -...

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Unformatted text preview: Probability Distributions, Continuous Variables Objectives: (Chapter 6, DeCoursey) - To establish the difference between probability distribution for discrete and continuous variables. - To learn how to calculate the probability that a random variable, X, will fall between the limits of "a" and "b". Probability Distributions, Continuous Variables A continuous variable, X, can take on an infinite number of values over a particular interval [a, b]. f(x) The probability that the variable X will lie between theses two endpoints is given by a b Pr[a x b] = f ( x)dx a b ([a, b] is sub-interval.) f(x) 0, f(x) is called probability density function. Probability Distributions, Continuous Variables f(x) Probability Distributions, Continuous Variables Cumulative Distribution: The probability that the continuous random variable is less than some upper value, call it x1, is given by x1 a b b Pr[ X x1 ] = [ ([a, b] is sub-interval.) Pr[a x b] = f ( x)dx a Compared with discrete variable: a xi b p( x ) i In most cases, we don't need to integrate all the way from -. If there is a lower value below which the f(x) is zero, we integrate from that lower value. Compared with the discrete variable: - f ( x)dx Pr[ X x] = xi x p( x ) i Probability Distributions, Continuous Variables Cumulative Distribution: + Probability Distributions, Continuous Variables Expected Value is the mean value of the distribution of the random variable. F ( ) = f(x) - f ( x)dx = 1 = E( X ) = + - xf ( x)dx i i Compared with discrete variable: E( X ) = x = all xi p( x ) = 1 i all xi ( x ) p( x ) 1 Probability Distributions, Continuous Variables Variance: + x 2 = E[( x - x ) 2 ] = ( x - x ) 2 f ( x)dx - + = - x 2 f ( x)dx - x 2 Standard Deviation:x Compared with discrete variable: x 2 = xi2 p ( xi ) - x2 all i 2 ...
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This note was uploaded on 02/13/2012 for the course MATH 2320 taught by Professor Glyn during the Fall '11 term at Minnesota.

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Probability distribution, continuous random variables -...

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