Statistics in geophysics probability theory II

Example: In an opinion poll, we might to decide to ask 50people whether they agree or disagree with a certain issue.The sample space for this experiment has 250 elements.Define a variabl

Statistics in Geophysics: Probability Theory II

Steffen Unkel

Department of Statistics Ludwig-Maximilians-University Munich, Germany

Random variables

variable than with the original probability structure

Example: In an opinion poll, we might to decide to ask 50people whether they agree or disagree with a certain issue.The sample space for this experiment has 250 elements.Define a variable X = number of 1s recorded out of 50.The sample space for X is the set of integers {0, 1, 2, , 50}

Random variables

We define an inducedprobability function PX on X as follows:

PX(X = xi) = P({ωj ∈ Ω : X (ωj) = xi}) ,for i = 1, , m and j = 1, , n

We will simply write P(X = xi) rather than PX(X = xi)

Example: Tossing a fair coin three times

X : number of heads obtained in the three tosses

Properties of a cdf

The function FX(x ) is a cdf if and only if the following threeconditions hold:

2 FX(x ) is a monotone, non-decreasing function of x

3 FX(x ) is continuous from the right; that is,

Density and mass functions

Definition:

A random variable X iscontinuous if FX(x ) is a continuousfunction of x A random variable isdiscrete if FX(x ) is a stepfunction of x

Associated with a random variable X and its cdf FX(x ) is anotherfunction, called either theprobability density function(pdf) or

probability mass function(pmf)

Probability mass function

Definition:

The probability mass function (pmf) of a discrete random variable

X is given by

fX(x ) = P(X = x ) for all x Hence, for positive integers a and b with a ≤ b, we have

Probability density function

A pmf gives us “point probabilities” and we can sum over thevalues of the pmf to get the cdf

The analogous procedure in the continuous case is to

substitute integrals for sums:

Probability density function

P(a < X < b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a ≤ X ≤ b)

Figure: Hypothetical pdf for a non-negative random variable X

Mean of a random variable

Definition:

denoted by E(X ) (or µX), is (provided that the sum or integralexists):

Expected value of a function of a random variable

Let g (x ) be a function of a random variable X Then (providedthat the sum or integral exists),

Properties of expected values

If X is any random variable, then (as long as the expectationsexist):

2 E(c g (X )) = cE(g (X ))

3 E(c1g1(X ) + c2g2(X )) = c1E(g1(X )) + c2E(g2(X ))

4 E(g1(X )) ≤ E(g2(X )) if g1(x ) ≤ g2(x ) for all x

Variance of a random variable

Linear transformations of random variables

Assume X is a random variable with mean µX and variance σX2 If

Y = aX + b, where a and b are any constants, then

µY = aµX + b, σY2 = a2σX2, σy = |a|σX

Statisticaldistributionsare used to model populations

We usually deal with a familyof distributions, which is

Here, we catalog someof the frequentlyoccurring probability

to their usage

This presentation is by no means comprehensive in its

coverage of statistical distributions!

Binomial distribution

Let X count the number of successes observed in a sequence

of n identical and independent Bernoulli trials, that is,

If X ∼ P(λ), then E(X ) = λ and Var(X ) = λ.

Example: Annual Hurricane Landfalls on the U.S coastline

Figure: Histogram of annual numbers of U.S landfalling hurricanes for

1899-1998 (dashed), and fitted Poisson distribution with λ = 1.7 (solid).

Approximation of the binomial distribution

Approximation of the binomial distribution

Exponential distribution

The exponential distribution can be used to model lifetimes

If X is a continuous random variable with non-negative range,which has pdf

fX(x ) = λ exp(−λx ) , for x ≥ 0 ,where λ > 0, then X is defined to have an exponential

distribution, denoted by X ∼ E (λ)

If X ∼ E (λ), then E(X ) = λ1 and Var(X ) = λ12

If the number of events in the unit time interval follows aPoisson distribution with mean λ, then the time to the next

If X is a normal random variable, then E(X ) = µ and

Var(X ) = σ2

Standard normal distribution

Normal distribution having µ = 0 and σ = 1:

Φ(z) =R−∞z φ(u)du = P(Z ≤ z) is the conventional notation forits cdf

Any Gaussian random variable can be standardized by subtractingits mean and dividing by its standard deviation:

Distributions of functions of a random variable

Let X be a continuous random variable with density fX(x ) andlet Y = g (X ) be astrongly monotone anddifferentiable

function

The density fY(y ) of Y is given by

fY(y ) = fX(g−1(y )) ·

dg−1(y )dy

for y > 0 and zero elsewhere.

If Y is a log-normal random variable, then

Vector of random variables

We need to know how to describe and use probability modelsthat deal with more than one random variable at a time(called multivariatemodels)

variables

A bivariate random vector (X , Y ) associates an ordered pair

of real numbers, that is, a point (x , y ), with each

Joint and marginal distributions

The two cases we will discuss are those in which (X,Y) isdiscrete or in which (X , Y ) is continuous

When (X , Y ) is discrete, thejoint pmfis

fX ,Y(x , y ) = P(X = x , Y = y ) ,where fX ,Y(x , y ) ≥ 0 for all (x , y ) and must sum to 1, if weadd over all possible observed vectors

The marginalpmfs of X and Y , fX(x ) = P(X = x ) and

fY(y ) = P(Y = y ), are given by

fX(x ) =XfX ,Y(x , y ) and fY(y ) =XfX ,Y(x , y )

Joint and marginal distributions

The joint cdf of two random variables, FX ,Y(x , y ) is

Example of a joint density

2

f(x, y)

0.05 0.10

0.15

Density of the bivariate standard normal distribution

fX |Y = fX ,Y(x , y )

fY(y ) .Conditional pmf and pdf are defined for any y such that fY(y ) > 0

Definition:

Let (X , Y ) be a bivariate random vector with joint pdf or pmf

fX ,Y(x , y ) and marginal pdfs or pmfs fX(x ) and fY(y ) Then Xand Y are calledindependentif, for every x , y ∈ R,

fX ,Y = fX(x )fY(y )

If X and Y are independent, then

fX |Y(x |y ) = fX(x ) and fY |X(y |x ) = fY(y )

Covariance and correlation

Thecovarianceof X and Y is the number defined by

Cov(X , Y ) = E[(X − µX)(Y − µY)] = E(XY ) − µXµY

Thecorrelationof X and Y is the number defined by

ρXY = Cov(X , Y )

The value ρXY is also called thecorrelation coefficient X and Yare calleduncorrelatedif ρXY = 0; they arepositively (negatively)correlated if ρXY > 0 (ρXY < 0)

Properties of covariance and correlation

The following statements hold:

Cov(X , Y ) = 0 and ρXY = 0

two constants, then

Var(aX + bY ) = a2Var(X ) + b2Var(Y ) + 2ab Cov(X , Y )

Var(aX + bY ) = a2Var(X ) + b2Var(Y )

Example: The bivariate standard normal distribution

The bivariate standard normal distribution with parameter ρ(|ρ| < 1) has the joint density

The correlation of X and Y is ρ

In this case: Uncorrelatedness implies independence

Example: The bivariate standard normal distribution

x

0.02 0.06

0.1 0.12

0.08 0.1

0.12 0.14

0.16 0.18

0.2 0.22

0.1 0.12

0.14 0.16

Sums of random variables

If X and Y are independent random variables with pmfs orpdfs fX(x ) and fY(y ), then the pmf or pdf of Z = X + Y is

if X and Y are continuous

The function fZ(z) is called the convolution of fX(x ) and

fY(y )

Law of large numbers

Considerindependently and identically distributed (i.i.d) randomvariables X1, X2, , Xn with E(Xi) = µ and Var(Xi) = σ2 < ∞(i = 1, , n)

it can be shown that E( ¯Xn) = µ and Var( ¯Xn) = 1nσ2

The law of large numbers states that

P

lim

n→∞

... real numbers, that is, a point (x , y ), with each

Joint and marginal distributions

The... distributions

The two cases we will discuss are those in which (X,Y) isdiscrete or in which (X , Y ) is continuous

When (X , Y ) is discrete, thejoint pmfis

fX ,Y(x , y )... class="text_page_counter">Trang 38

Joint and marginal distributions

The joint cdf of two random variables, FX ,Y(x ,