2 random variables
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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.
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Distribution of a random variableDistribution function*RANDOM VARIABLES
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Distribution function properties*RANDOM VARIABLES
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*RANDOM VARIABLESFor a random variable, we define
Probability function
Density function,
depending on wether is either discrete or continuous
Distribution of a random variable
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Probability function *verifiesRANDOM VARIABLESDistribution of a random variable
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Probability density function *verifiesWe haveRANDOM VARIABLESDistribution of a random variable
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completely determines the distributionof a random variable.*RANDOM VARIABLESDistribution of a random variable
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Discrete random variablesPoint mass *RANDOM VARIABLES
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Discrete uniform*RANDOM VARIABLESDiscrete random variables
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Bernoulli*RANDOM VARIABLESDiscrete random variables
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BinomialSuccesses in n independent Bernoulli trials with success probability p *RANDOM VARIABLESDiscrete random variables
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Geometric
Time of first success in a sequence of independent Bernoulli trials with success probability p *RANDOM VARIABLESDiscrete random variables
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Poisson
X expresses the number of rare events*RANDOM VARIABLESDiscrete random variables
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Uniform*RANDOM VARIABLESContinuous random variables
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Exponential*RANDOM VARIABLESContinuous random variables
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Normal*RANDOM VARIABLESContinuous random variables
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Properties of normal distribution
standard normal
(ii)
(iii) independent i=1,2,...,n*RANDOM VARIABLESContinuous random variables
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Transformations of random variablesX random variable with ;
Y = r(x); distribution of Y ?
r() is one-to-one; r -1().*RANDOM VARIABLES
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(X,Y) random variables;
If (X,Y) is a discrete random variable
If (X,Y) is continuous random variable*RANDOM VARIABLESBivariate random variables
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The marginal probability functions for X and Y are:
*RANDOM VARIABLESBivariate random variablesFor continuous random variables, the marginaldensities for X and Y are:
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Independent random variablesTwo random variables X and Y are independent ifand only if:
for all values x and y.*RANDOM VARIABLES
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Conditional distributionsDiscrete variables*If X and Y are independent:Continuous variablesRANDOM VARIABLES
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Expectation of a random variable*Properties:
(i)
If are independent then:RANDOM VARIABLES
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Moment of order k*RANDOM VARIABLES
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VarianceGiven X with :
standard deviation*RANDOM VARIABLES
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VarianceProperties:
(i)
If are independent then
(iii)
(iv)*RANDOM VARIABLES
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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)
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Correlation*RANDOM VARIABLESX and Y random variables;
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*RANDOM VARIABLESCorrelationProperties
(i)
If X and Y are independent then
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Expectation of transformed variables *RANDOM VARIABLES
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Sample mean and sample variance*Sample meanSample varianceRANDOM VARIABLES
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Properties
X random variable; i. i. d. sample,
Then:
(i)
(ii)
(iii)*RANDOM VARIABLESSample mean and sample variance
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Conditional expectationX and Y are random variables;Then:*Properties:RANDOM VARIABLES
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