2. Random variables Introduction Distribution of a random variable Distribution function properties Discrete random variables Point mass Discrete uniform Bernoulli Binomial Geometric Poisson

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2. Random variables Continuous random variables Uniform Exponential Normal Transformations of random variables Bivariate random variables Independent random variables Conditional distributions Expectation of a random variable kth moment

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2. Random variables Variance Covariance Correlation Expectation of transformed variables Sample mean and sample variance Conditional expectation

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RANDOM VARIABLESIntroductionRandom variables assign a real number to eachoutcome:

*Random variables can be:

Discrete: if it takes at most countably many values (integers). Continuous: if it can take any real number.

Distribution of a random variableDistribution function*RANDOM VARIABLES

Distribution function properties*RANDOM VARIABLES

*RANDOM VARIABLESFor a random variable, we define

Probability function

Density function,

depending on wether is either discrete or continuous

Distribution of a random variable

Probability function *verifiesRANDOM VARIABLESDistribution of a random variable

Probability density function *verifiesWe haveRANDOM VARIABLESDistribution of a random variable

completely determines the distributionof a random variable.*RANDOM VARIABLESDistribution of a random variable

Discrete random variablesPoint mass *RANDOM VARIABLES

Discrete uniform*RANDOM VARIABLESDiscrete random variables

Bernoulli*RANDOM VARIABLESDiscrete random variables

BinomialSuccesses in n independent Bernoulli trials with success probability p *RANDOM VARIABLESDiscrete random variables

Geometric

Time of first success in a sequence of independent Bernoulli trials with success probability p *RANDOM VARIABLESDiscrete random variables

Poisson

X expresses the number of rare events*RANDOM VARIABLESDiscrete random variables

Uniform*RANDOM VARIABLESContinuous random variables

Exponential*RANDOM VARIABLESContinuous random variables

Normal*RANDOM VARIABLESContinuous random variables

Properties of normal distribution

standard normal

(ii)

(iii) independent i=1,2,...,n*RANDOM VARIABLESContinuous random variables

Transformations of random variablesX random variable with ;

Y = r(x); distribution of Y ?

r() is one-to-one; r -1().*RANDOM VARIABLES

(X,Y) random variables;

If (X,Y) is a discrete random variable

If (X,Y) is continuous random variable*RANDOM VARIABLESBivariate random variables

The marginal probability functions for X and Y are:

*RANDOM VARIABLESBivariate random variablesFor continuous random variables, the marginaldensities for X and Y are:

Independent random variablesTwo random variables X and Y are independent ifand only if:

for all values x and y.*RANDOM VARIABLES

Conditional distributionsDiscrete variables*If X and Y are independent:Continuous variablesRANDOM VARIABLES

Expectation of a random variable*Properties:

(i)

If are independent then:RANDOM VARIABLES

Moment of order k*RANDOM VARIABLES

VarianceGiven X with :

standard deviation*RANDOM VARIABLES

VarianceProperties:

(i)

If are independent then

(iii)

(iv)*RANDOM VARIABLES

CovarianceX and Y random variables;*RANDOM VARIABLES Properties

(i) If X, Y are independent then

(ii)

(iii) V(X + Y) = V(X) + V(Y) + 2cov(X,Y)

V(X - Y) = V(X) + V(Y) - 2cov(X,Y)

Correlation*RANDOM VARIABLESX and Y random variables;

*RANDOM VARIABLESCorrelationProperties

(i)

If X and Y are independent then

Expectation of transformed variables *RANDOM VARIABLES

Sample mean and sample variance*Sample meanSample varianceRANDOM VARIABLES

Properties

X random variable; i. i. d. sample,

Then:

(i)

(ii)

(iii)*RANDOM VARIABLESSample mean and sample variance

Conditional expectationX and Y are random variables;Then:*Properties:RANDOM VARIABLES