Correlation and Dependence - Bivariate Normal Distribution

Bivariate Normal Distribution

If a pair (X, Y) of random variables follows a bivariate normal distribution, the conditional mean E(X|Y) is a linear function of Y, and the conditional mean E(Y|X) is a linear function of X. The correlation coefficient r between X and Y, along with the marginal means and variances of X and Y, determines this linear relationship:

E(Y|X) = E(Y) + r\sigma_y\frac{X-E(X)}{\sigma_x},

where E(X) and E(Y) are the expected values of X and Y, respectively, and σx and σy are the standard deviations of X and Y, respectively.

Read more about this topic:  Correlation And Dependence

Famous quotes containing the words normal and/or distribution:

    “Literature is a defense against the attacks of life. It says to life: “You can’t deceive me. I know your habits, foresee and enjoy watching all your reactions, and steal your secret by involving you in cunning obstructions that halt your normal flow.””
    —Cesare Pavese (1908–1950)

    “My topic for Army reunions ... this summer: How to prepare for war in time of peace. Not by fortifications, by navies, or by standing armies. But by policies which will add to the happiness and the comfort of all our people and which will tend to the distribution of intelligence [and] wealth equally among all. Our strength is a contented and intelligent community.”
    —Rutherford Birchard Hayes (1822–1893)