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Random Vector

Random Vector. Martina Litschmannová m artina.litschmannova @vsb.cz K210. Random Vectors. An k - dimensional random vector is a function X = that associates a vector of real numbers with each element i s a random variable . For example :

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Random Vector

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  1. RandomVector Martina Litschmannová martina.litschmannova@vsb.cz K210

  2. Random Vectors • An k- dimensional random vector is a function X = that associates a vector of real numbers with each element is a randomvariable. Forexample: • A semiconductor chip is divided into ‘M’ regions. For the random experiment of finding the number of defects and their locations, let denote the number of defects in ith region. Then is a discreterandomvector. • In a random experiment of selecting a student’s name, let = height of ithstudent in inches and = weight of ithstudent in pounds.Thenis a continuous randomvector.

  3. 2 – dimensionalRandomVectors • We'regoing to focus on 2-dimensional distributions (i.e. randomvectorconsistsonly oftwo random variables) but higher dimensions (more than two variables) are also possible.

  4. Joint Probability Distribution • Jointdistribution for randomvectordefines the probability of events defined in terms of both X and Y. Joint cumulativedistributionfunctionforisgiven by .

  5. Joint cumulativedistributionfunction Joint cumulativedistributionfunctionforisgiven by . Propertiesof joint CDF: , , , isnondecreasing in eachvariable, iscontinousfromtheleft in eachvariable.

  6. Discrete Joint Probability Distributions The probability function, also known as the probability mass function for a joint probability distributionis defined such that: .

  7. Discrete Joint Probability Distributions ProbabilityMass Function for a JointProbabilityDistribution: • Properties ofp.m.f.: • only fora finite or countable set of values • , • , • IfX, Y are independent Table of joint probabilities

  8. A random experiment consists of tossing coin (X)and flippingdie (Y). Find probabilitymass function for a jointprobabilitydistributionof a randomvector.

  9. A random experiment consists of tossing coin (X)and flippingdie (Y). Find probabilitymass function for a jointprobabilitydistributionof a randomvector.

  10. A random experiment consists of tossing coin (X)and flippingdie (Y). Find probabilitymass function for a jointprobabilitydistributionof a randomvector. control cell

  11. Probabilitymass function for a jointprobabilitydistributionof a randomvectorisgiven as: • Find:

  12. ContinousJoint Probability Distributions , where is Joint Probability Density Function. Propertiesof Joint Probability DensityFunction: • , • , • If exists, pak , • .

  13. Find theconstantc so thatfunctioncanbe a joint probability densityfunctionof a randomvector. That thefunctioncanbe a joint probability densityfunctionof a randomvector, i condition .

  14. Marginal probability distributions Obtained by summing or integrating the joint probability distribution over the values of the other random variable. • DiscreteRandomVector , . • ContinousRandomVector , , .

  15. A random experiment consists of tossing coin (X)and flippingdie (Y). Probabilitymass function for a jointprobabilitydistributionof a randomvector Find Marginal Probability MassFunctions and

  16. A random experiment consists of tossing coin (X)and flippingdie (Y). Probabilitymass function for a jointprobabilitydistributionof a randomvector Find Marginal Probability MassFunctions and

  17. A random experiment consists of tossing coin (X)and flippingdie (Y). Probabilitymass function for a jointprobabilitydistributionof a randomvector Marginal Probability MassFunctions and

  18. A random experiment consists of tossing coin (X)and flippingdie (Y). Probabilitymass function for a jointprobabilitydistributionof a randomvector Marginal Probability MassFunctions and

  19. A certain farm produces two kinds of eggs on any given day; organic and non-organic. Let these two kinds of eggs be represented by the random variables X and Y respectively. Given that the joint probability density function of these variables is given byFind:a) marginaldensityfunctionsand

  20. A certain farm produces two kinds of eggs on any given day; organic and non-organic. Let these two kinds of eggs be represented by the random variables X and Y respectively. Given that the joint probability density function of these variables is given byFind:b)

  21. A certain farm produces two kinds of eggs on any given day; organic and non-organic. Let these two kinds of eggs be represented by the random variables X and Y respectively. Given that the joint probability density function of these variables is given byFind:c)

  22. Conditionalprobability distributions Conditional Probability Distributions arise from joint probability distributions where by we need to know that probability of one event given that the other event has happened, and the random variables behind these events are joint. • DiscreteRandomVector 0, . • ContinousRandomVector 0, .

  23. Joint and marginal probabilitydistributionof a randomvectorisgiven as: • Find:

  24. Jointprobability density function of is given byFind:a) conditionaldensityfunction

  25. Jointprobability density function of is given byFind:b) conditionaldensityfunction

  26. Conditionalexpectedvalue (expectation) Conditional expectationis the expected value of a real random variable with respect to a conditional probability distribution. Discreterandomvector: Continousrandomvector:

  27. Conditional variance Conditionalvariance is the variance of a conditional probability distribution.

  28. Joint and marginal probabilitydistributionof a randomvectorisgiven as: • Find:

  29. Jointprobability density function of is given byFind:a) Eb) D.

  30. Independence Two random variables X and Yare independent if • DiscreteRandomVariables • ContinousRandomVector .

  31. MeasuresofLinear Independence Covariance: Correlationcoefficient: • is a scaled version of covariance

  32. Covariance Covariance: Covariance matrix:

  33. Correlation Correlation: • … are positivelycorrelated • … are negatively correlated • … are uncorrelated Correlation matrix:

  34. Correlation =1,000 = -1,000 =0,000 =0,934 =0,967 =-0,143 =0,857 =0,608

  35. Joint and marginal probabilitydistributionof a randomvectorisgiven as: • Find: • , • , • , • Are randomvariableX, Y independent? • Are randomvariableX, Ylinear independent?

  36. Joint probability density function of is given byMarginaldensityfunctions are: • Find: • , • , • , • Are randomvariableX, Y independent? • Are randomvariableX, Ylinear independent?

  37. Study materials : • http://homel.vsb.cz/~bri10/Teaching/Bris%20Prob%20&%20Stat.pdf (p. 64 - p.70)

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