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Continuous Random Variables: Summary of The Uniform and Exponential Probability Distributions

Module by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

Summary: This module provides a summary of formulas and definitions related to Continuous Random Variables. Note: This module is currently under revision, and its content is subject to change. This module is being prepared as part of a statistics textbook that will be available for the Fall 2008 semester.

Note: You are viewing an old version of this document. The latest version is available here.

formula 1: Uniform

XX = a real number between aa and bb (in some instances, XX can take on the values aa and bb) aa = smallest XX ; bb = largest XX

X~U(a, b)X~U(a,b)

The mean is μ=a+b 2 μ a+b 2

The standard deviation is σ= (b-a)2 12 σ (b-a)2 12

Probability density function: fX = 1b-a fX=1b-a for a<X<b a X b or aXb a X b

Area to the Left, Right and Inbetween: P(X<x)=(a base)(height) P(X x) (a base)(height)

P(X>x)=(a base)(height) P(X x) (a base)(height)

P(c<X<d)=(a base)(height)=(d-c)(height) P(c X d) (a base)(height) (d-c)(height).

formula 2: Exponential

XX = a real number, 0 or larger mm = the parameter that controls the rate of decay or decline

X~Exp (m)X~Exp(m)

The mean an standard deviation are the same.

The probability density function: f(X) =m e-m⋅Xf(X)=me-m⋅X

μ=σ= 1mμ=σ=1m and m= 1μ= 1σm=1μ=1σ

Area to the Left: P(X<x)=1- e-m⋅x P(X x) 1- e-m⋅x

Area to the Right: P(X>x)= e-m⋅x P(X x) e-m⋅x

Area Inbetween: P ( c < X < d ) = P ( X < d ) - P ( X < c ) = ( 1 - e − m⋅d ) - ( 1 - e − m⋅c ) = e − m⋅c - e − m⋅d P ( c X d ) = P ( X d ) - P ( X c ) =(1- e − m⋅d )-(1- e − m⋅c )= e − m⋅c - e − m⋅d

Percentile, k: k = LN(1-AreaToTheLeft) -m k= LN(1-AreaToTheLeft) -m

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