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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 subinterval.) 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 subinterval.) 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.
 Fall '11
 Glyn
 Math, Probability

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