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TWO DIMENSIONAL RANDOM VARIABLES

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CONTENTS UNIT – II TWO DIMENSIONAL RANDOM VARIABLES CHAPTER – I • Definition of Two dimensional random variables • Joint Distributions Distribution function Probability mass functio ... n Probability density function • Marginal and Conditional Distributions CHAPTER – II • Covariance • Correlation • Regression CHAPTER – III • Transformation of Random VariablesTWO DIMENSIONAL RANDOM VARIABLES 2 UNIT – III TWO DIMENSIONAL RANDOM VARIABLES CHAPTER – I INTRODUCTION TWO DIMENSIONAL RANDOM VARIABLES In the last unit, we introduced the concept of a single random variable.We observed that the various statistical averages or moments of the random variable like mean, variance, standard deviation, skewness give an idea about the characteristics of the random variable.But in many practical problems several random variables interact with each other and frequently we are interested in the joint behaviour of these random variables. For example, to know the health condition of a person, doctors measure many parameters like height, weight, blood pressure, sugar level etc. We should now introduce techniques that help us to determine the joint statistical properties of several random variables.The concepts like distribution function, density function and moments that we defined for single random variable can be extended to multiple random variables also. Two Dimensional Random Variable Let S be the sample space associated with a random experiment E.Let X=X(s) and Y=Y(s) be two functions each assigning a real number to each outcomes s Є S. Then (X , Y) is called a two dimensional random variable. Note (i)If the possible values of (X , Y) are finite or countable infinite, (X ,Y) is called a two dimensional discrete random variable. (ii)If (X , Y) can assume all values in a specified region R in the xy-plane, (X ,Y) is called a two dimensional continuous random variable.TWO DIMENSIONAL RANDOM VARIABLES 3 Joint probability Mass Function (Discrete Case) If (X,Y) is a two-dimensional discrete RV such that P(X = xi , Y = yj) = pij , then pij is called the probability mass function of (X , Y) provided pij ≥ 0 for all i and j. (ii) p 1 ∑∑ ij = i j Joint probability Density Function (Continuous Case) If (X,Y) is a two-dimensional continuous RV such that x- ( , ) 2 2 2 2 , f(x,y) is called the joint pdf of (X,Y),provided f(x,y) satisfies the following conditions. dx dx dy dy p X x and y y f x y dx dy then     ≤ ≤ + − ≤ + =   (i) f (x , y) ≥ 0, for all (x , y) ∈ R, where R is the range space ( ) ( , ) d y = 1 . R i i f x y d x ∫∫ Note {( ) D}= ( , ) . In particular D P X Y f x y dx dy < ∈ ∫∫ { , } ( , ) d b c a P a X b c Y d f x y dx dy ≤ ≤ ≤ ≤ = ∫∫ Cumulative Distribution Function If ( X , Y) is a two-dimensional RV (discrete or continuous ), then F(x , y) = P{ X ≤ x and Y ≤ y } is called the cdf of (X,Y). In the discrete case,TWO DIMENSIONAL RANDOM VARIABLES 4 ( , ) = ∑∑ j i F x y pij In the continuous case, ( , ) ( , ) dy y x F x y f x y dx −∞ −∞ = ∫ ∫ Properties of F (x , y) [Show More]

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