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This module provides a summary of formulas and definitions related to Continuous Random Variables.
Formula

Uniform

X = a real number between a and b (in some instances, X can take on the values a and b ). a = smallest X ; b = largest X

X ~ U ( a, b )

The mean is μ a + b 2

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

Probability density function: f X = 1 b - a for a X b

Area to the Left of x: P ( X x ) (base) (height)

Area to the Right of x: P ( X x ) (base) (height)

Area Between c and d: P ( c X d ) ( base ) ( height ) ( d - c ) ( height ) .

Formula

Exponential

X ~ Exp ( m )

X = a real number, 0 or larger. m = the parameter that controls the rate of decay or decline

The mean and standard deviation are the same.

μ = σ = 1 m and m = 1 μ = 1 σ

The probability density function: f ( X ) = m e -m⋅X , X 0

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

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

Area Between c and 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

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Source:  OpenStax, Collaborative statistics. OpenStax CNX. Jul 03, 2012 Download for free at http://cnx.org/content/col10522/1.40
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