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Unformatted text preview: : Conditional density. Let ( , ) be a two dimensional random variable with joint density and marginal densities and . Then the conditional density for given is denoted and defined by XY X Y Def X Y f f f X Y y =   ( , ) , ( ) ( ) similarly, ( , ) , ( ) ( ) XY X y Y Y XY Y x X X f x y f f y f y f x y f f x f x = = Simple Linear Regression and Correlation Curve Of Regression  Let ( , ) be a 2dimensional random variable, (1) The graph of the mean value of given , denoted by is called the curve of regression of on . (2) The graph of the mean value of X y X Y X Y y X Y Y μ =  given , denoted by is called the curve of regression of on . Y x X x Y X μ = We shall now assume that is not a random variable, rather , it is a mathematical variableAn entity that can assume different values but whose values at the time of consideration is not determined by X chance. Example: Suppose we are developing a model to describe the temperature of the water off the continental shelf. Since temperature depends in part on the depth of the water, 2 variables are involved; X=depth, Y=temperature. We are not interested in making inferences on the depth of the water. With X(depth) fixed, the temperature measurements(Y) at different places (of depth X) varies.   Here we have a conditional random variable denoted by  ,this conditional random variable has a mean , which is a function of . The graph of is called the curve of regression of on , he x Y x Y x Y x Y X μ μ re depends on and is called the or variable. The variable whose value is used to help predict the behaviour of  is called the or variable or the x Y X dependent response X Y independent predictor regre . Sometimes the values of used can be preselected and in this case the study is said to be , otherwise it is called study. ssor X controlled observational  1 1 1 1 Linear Curve of Regression of on is given by , denotes the intercept and denotes the slope of the regression line. We need to evaluate and . Y x Y X x μ β β β β β β β β = + ∈ ¡ 1 2 1 2 Let , ,..., be values of (these points are assumed to be measured without error) We are concerned with the random variables,  ,  ,...,  . A random variable varies abou n n x x x x x x n X n Y Y Y   t its mean value. Let   i x i i x i i x Y x i Y E Y Y E μ μ = ⇒ = + x i Y 1 1 We assume that the random difference has mean 0. Since we are assuming that the regression is linear we can conclude that Therefore, Simple Linear Regressi i i i i i E x Y x E μ β β β β = + = + + → on Model where where is assumed to be a random variable with mean 0. i i x i Y Y E = So now we have a data consisting of a collection of pairs ( , ), where is an observed value of the variable and is the corresponding observation for the random variable . The observed valu i i i i n x y x X y Y 1 e usually differs from its mean value by some random amount....
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This note was uploaded on 02/23/2011 for the course MATH 112 taught by Professor Ritadubey during the Spring '11 term at Amity University.
 Spring '11
 ritadubey
 Correlation, Linear Regression

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