Reg n Corr - : Conditional density. Let ( , ) be a two...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 2-dimensional 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 = | given , denoted by is called the curve of regression of on . Y x X x Y X = We shall now assume that is not a random variable, rather , it is a mathematical variable-An 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 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 = + 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 =- = + 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 = + = + + 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....
View Full Document

Page1 / 65

Reg n Corr - : Conditional density. Let ( , ) be a two...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online