4.1-4.3 - Chapter 4 Continuous Distributions Definition: A...

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Chapter 4 Continuous Distributions Definition: A function f(x) is called the density of a continuous variable X if (i) ; 0 ) ( x f (ii) 1 ) ( = - dx x f (a) The graph of ) ( x f is called “density curve”. (b) Proportion of X - values between ‘a’ and ‘b’=P( b X a ) = ) ( b a dx x f = area of density curve between a and b (c ) Proportion of X - values with b X a = Proportion of x with b X a < < . ( Since the area of the curve under single value = 0, P(X=a)=0 for any constant a). 1
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Examples of the Graphs of Continuous Distributions: 1. Sketch the graph of uniform distribution and give the formula for the m ass function f(x) a. on the interval [0,1] b. on [a,b] 2. Exponential distribution x 0 >0 ( ) 0 x<0 x e f x λ - = The Graph: Verify that it is a legitimate distribution. (Is ( ) 1 f x f - = ?) How can we find a proportion of all values X that fall between two given numbers? Answer: . ... 2
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Example 1 : Let X denotes the amount of time ( hr ) the music is played in a program of a radio station. A potential sponsor wants to study the distribution of x. Suppose, the density function is ) ( x f = 90 x 8 (1- x ), 0 ≤ x 1 0, otherwise. (i) Note ) ( x f is a density. (ii) Proportion of programs between 0.7 and 0.9 (hr) is . 587 . 0 ) ( 9 . 7 . = dx x f (iii) The constant c such that .5 ) ( ) ( 1 = = c c O dx x f dx x f 3
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.5 10 9 90 10 9 = - c c Solving by a numerical procedure, c 2245 .838 50% of programs have music < .838 hrs. HW assignment: Section 4.1: #1-9 odd, 8* 4
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4.2 Cumulative distribution functions and expected Values Cumulative distribution(cdf): Let f(x) be the density of a random variable X then F(x)=P(X x)= - x dt t f ) ( is called the cumulative distribution function of X. Example. If f(x)=2x, 0<x<1, f(x)=0 otherwise then F(x)= - x dt t f ) ( = 2 0 2 x tdt x = for 0<x<1, F(x)=1, for x 1 and F(x)=0, for x . 0 Using F(x) to calculate probabilities. We can use F to calculate probabilities, P(a X b )=F(b)-F(a). P(X>0.5)=1-F(0.5)……
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This note was uploaded on 07/25/2008 for the course STT 430 taught by Professor Nane during the Spring '08 term at Michigan State University.

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4.1-4.3 - Chapter 4 Continuous Distributions Definition: A...

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