6125.pdf

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Lecture 6

Continuous Probability Distributions

Statistics for

Civil & Environmental Engineers

Uniform Distribution(1)

Uniform pdf

Definition

A uniformly distributed random variate can have any value in an interval ato bwith equal likelihood

Population parameters

Mean=

Variance=

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Uniform Distribution(2)

Parameter estimators

MOM:

MLE:

Distribution shapes

Statistics for Civil & Environmental Engineers

Uniform Distribution Example

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Exponential Distribution(1)

Definition

The exponential distribution models the time (or length or area) between Poisson events

Hann(1994). Statistical methods in hydrology

Exponential Distribution(2)

Exponential pdf & cdf

pdf:

cdf:

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Mean=

Variance=

Skewness coef.= 2

Parameter estimator Population parameters

Comments

special case of the gamma distribution (r =1)

Distribution shapes

pdf cdf λ

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Exponential Distribution Example

Gamma Distribution(1)

Description

the probability dist of the time to the r th occurrence, which is the sum of rindependent r.v. T₁+ T₂+...+ T_rform the exponential distribution

Gamma pdf

1

( ; , ) for 0 with 0 and >0,

( )

0 otherwise

r r x

X

f x r x e x r

r λ

λ

λ λ

− −

= ≤ >

Γ

=

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Mean=

Variance=

Skewness coef.=

Population parameters

Parameter estimators

Comments

the log-Pearson type III distribution is a 3 parameter gamma distribution

Distribution shapes

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Gamma Distribution Example

Normal Distribution(1)

Exponential pdf & cdf

pdf:

cdf:

Comments

2 parameter distribution

skewness coef.=0 i.e. symmetrical about the mean

unbounded but if µ is greater than 3σ, the chances of Xless than 0 are negligible in practice

reproductive properties

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Standard Normal Distribution

Central Limit Theorem

If S_n is the sum ofnindependently and identically distributed random variables X_ieach having a mean µ and variance σ²then in the limit as napproaches infinity, the distribution of S_n

approaches a normal distribution with mean nµ and variance nσ²

Normal Distribution Example

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Central Limit Theorem

If S_n is the sum ofnindependently and identically distributed random variables X_ieach having a mean µ and variance σ²then in the limit as napproaches infinity, the distribution of S_n

approaches a normal distribution with mean nµ and variance nσ²

Normal Distribution Example(2)

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(2-parameter) Lognormal Distribution

Random variables

X: lognormally distribution Y=LN(X): normally distributed

Features

Population parameters

Bounded

Lognormal Distribution Example(1)

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Lognormal Distribution Example(2)