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

Module by: Susan Dean, Barbara Illowsky, Ph.D.

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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

XX ~ U(a, b)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

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

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

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

Formula 2: Exponential

XX ~ Exp (m)Exp(m)

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

The mean and standard deviation are the same.

μ=σ= 1mμ=σ=1m and m= 1μ= 1σm=1μ=1σ

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

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

Area to the Right of x: P(X>x)= e-m⋅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 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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