lecture 31

RecapRandom variables

Continuous random variable • Sample space has infinitely many elements

• The density function f(x) is a continuous function

• Calculation of probabilities

P( a < X < b ) = f(t) dt

Discrete random variable

• Sample space is finite or countably many elements

• The probability function f(x)Is often tabulated

• Calculation of probabilities

P( a < X < b) = f(t) a<t<b

b

a

lecture 32

Mean / Expected value Definition

Definition: Let X be a random variable with probability /Density function f(x). The mean or expected value of X is give by

if X is discrete, and

if X is continuous.

x

)x(fx)X(Eμ

dx)x(fx)X(Eμ

lecture 33

Mean / Expected value Interpretation

Interpretation:The total contribution of a value multiplied by the probability of the value – a weighted average.

Mean value= 1,5

x0 1 2 3

0.10.20.30.4

f(x)

Example:

lecture 34

Mean / Expected valueExample

Problem: • A private pilot wishes to insure his plane valued at 1 mill kr.• The insurance company expects a loss with the following probabilities:

• Total loss with probability 0.001• 50% loss with probability 0.01• 25% loss with probability 0.1

1. What is the expected loss in kroner ?2. What premium should the insurance company

ask if they want an expected profit of 3000 kr ?

lecture 35

Theorem: Let X be a random variable with probability / density function f(x). The expected value of g(X) is

if X is discrete, and

if X is continuous.

Mean / Expected value Function of a random variable

x

)X(g)x(f)x(g)]X(g[Eμ

dx)x(f)x(g)]X(g[Eμ)X(g

lecture 36

Theorem: Linear combinationLet X be a random variable (discrete or continuous), and let a and b be constants. For the random variable aX + b we have

E(aX+b) = aE(X)+b

Expected valueLinear combination

lecture 37

Mean / Expected valueExample

Problem: • The pilot from before buys a new plane valued at 2 mill kr.• The insurance company’s expected losses are unchanged:

• Total loss with probability 0.001• 50% loss with probability 0.01• 25% loss with probability 0.1

1. What is the expected loss for the new plane?

lecture 38

Mean / Expected valueFunction of a random variables

Definition:Let X and Y be random variables with joint probability / density function f(x,y). The expected value of g(X,Y) is

if X and Y are discrete, and

if X and Y are continuous.

x y

)Y,X(g)y,x(f)y,x(g)]Y,X(g[Eμ

dydx)y,x(f)y,x(g)]Y,X(g[Eμ)Y,X(g

lecture 39

Problem:Burger King sells both via “drive-in” and “walk-in”.Let X and Y be the fractions of the opening hours that “drive-in” and “walk-in” are busy.Assume that the joint density for X and Y are given by

Mean / Expected valueFunction of two random variables

4xy 0 x 1 , 0 y 10 otherwise

The turn over g(X,Y) on a single day is given by

g(X,Y) = 6000 X + 9000Y

What is the expected turn over on a single day?

f(x,y) = {

lecture 310

Mean / Expected value Sums and products

Theorem: Sum/ProductLet X and Y be random variables then

E[X+Y] = E[X] + E[Y]

If X and Y are independent then

E[X Y] = E[X] E[Y] ..

lecture 311

VarianceDefinition

Definition:Let X be a random variable with probability / density function f(x) and expected value . The variance of X is then given

if X is discrete, and

if X is continuous.

The standard deviation is the positive root of the variance:

x

222 )x(f)μx(])μX[(E)X(Varσ

dx)x(f)μx(])μX[(E)X(Varσ 222

)X(Varσ

lecture 312

VarianceInterpretation

Varians = 0.5 Varians = 2

The variance expresses, how dispersed the density / probability function is around the mean.

x 1 2 3

0.10.20.30.4

f(x)

x0 1 2 3 4

0.10.20.30.4

f(x)0.5 0.5

Rewrite of the variance:222 μ]X[E)X(Varσ

lecture 313

Theorem: Linear combinationLet X be a random variable, and let a be b constants. For the random variable aX + b the variance is

VarianceLinear combinations

Examples: Var (X + 7) = Var (X) Var (-X ) = Var (X) Var ( 2X ) = 4 Var (X)

)X(Vara)baX(Var 2

lecture 314

CovarianceDefinition

Definition:Let X and Y be to random variables with joint probability / density function f(x,y). The covariance between X and Y is

if X and Y are discrete, and

if X and Y are continuous.

x y

YXYXXY)y,x(f)μy)(μx()]μY)(μX[(E)Y,X(Covσ

dydx)y,x(f)μy)(μx()Y,X(CovσYXXY

lecture 315

CovarianceInterpretation

Covariance between X and Y expresses how X and Y influence each other.

Examples: Covariance between

• X = sale of bicycle and Y = bicycle pumps is positive.

• X = Trips booked to Spain and Y = outdoor temperature is negative.

• X = # eyes on red dice and Y = # eyes on the green dice is zero.

lecture 316

Theorem:The covariance between two random variables X and Y with means X and Y, respectively, is

CovarianceProperties

Notice! Cov (X,X) = Var (X)

If X and Y are independent random variables, then

Cov (X,Y) = 0

Notice! Cov(X,Y) = 0 does not imply independence!

YXXYμμ]YX[E)Y,X(Covσ

lecture 317

Variance/CovariaceLinear combinations

Theorem: Linear combinationLet X and Y be random variables, and let a and b beconstants.For the random variables aX + bY the variance is

Specielt: Var[X+Y] = Var[X] + Var[Y] +2Cov (X,Y)

If X and Y are independent, the variance is

Var[X+Y] = Var[X] + Var[Y]

)Y,X(Covba2)Y(Varb)X(Vara)bYaX(Var 22

lecture 318

CorrelationDefinition

Definition:Let X and Y be two random variables with covariance Cov (X,Y) and standard deviations X and Y, respectively.

The correlation coefficient of X and Y is

It holds that

If X and Y are independent, then

YX

XY σσ

)Y,X(Covρ

1ρ1XY

0ρXY

lecture 319

Mean, variance, covariaceCollection of rules

Sums and multiplications of constants:

E (aX) = a E(X) Var(aX) = a2Var (X) Cov(aX,bY) = abCov(X,Y)

E (aX+b) = aE(X)+b Var(aX+b) = a2 Var (X)

Sum:E (X+Y) = E(X) + E(Y) Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)

X and Y are independent: E(XY) = E(X) E(Y) Var(X+Y) = Var(X) + Var(Y)