2013LecturesPart4-CdfandMultivariateTransformations TopicsinStatistics

Topics in Statistics

2013 Lectures

Part 4 - Cdf and Multivariate Transformations

Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

random variable is a measurable real-valued function on subsets of sample space

distribution of a random variable is an assignment of probabilities to events{X ∈A},A⊂R

cumulative distribution function

F_X(t) =P(X ≤t),t ∈R

Theorem T8:For each random variableX, cdfF_X has the following properties:

a) F_X is nondecreasing;

b) limt→−∞F_X(t) =0, limt→∞F_X(t) =1;

c) F_X is continuous on the right.

Multivariate Transformations

To determine densities of bivariate/multivariate transformations is typically regarded as challenging.

Follow the steps:

a) To a transformation functionϕchoose a “companion function”ηsuch that

z =ϕ(x,y) w =η(x,y)

can be solved asx =α(z,w)andy =β(z,w).

b) Determine the imageDof the supportC of densityf in (z,w)-plane under the transformation.

c) Find the Jacobian of the transformation, i.e. the determinant

J =

∂α

∂z

∂α

∂β ∂w

∂z

∂β

∂w

d) Determine the joint density of(Z,W),

Z =ϕ(X,Y),W =η(X,Y)by

g(z,w) =

(f(α(z,w), β(z,w))|J| for(z,w)∈D,

0 otherwise.

e) Compute the density ofZ as the marginal density of(Z,W)

g_Z(z) = Z ∞

−∞

g(z,w)dw = Z

DZ

f(α(z,w), β(z,w))|J|dw, whereD ={w|(z,w)∈D}.

Multivariate Transformations

Example T14: (sum of random variables)

Determine the general density function of the sumZ =X +Y of continuous random variablesX andY.

Example T15: (triangular distribution)

Determine the distribution ofZ =X +Y whenX,Y ∼U(0,1).

Example T16: (product of random variables)

Determine the general density function of the productXY of continuous random variablesX andY.

f(x) = 1

√

2πσexp

−(x −µ)² 2σ²

is callednormalwith parametersµandσ >0.

Theorem T9:

Z ∞

−∞

√1

2πσexp

−(x−µ)² 2σ²

dx =1.