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:

    “As blacks, we need not be afraid that encouraging moral development, a conscience and guilt will prevent social action. Black children without the ability to feel a normal amount of guilt will victimize their parents, relatives and community first. They are unlikely to be involved in social action to improve the black community. Their self-centered personalities will cause them to look out for themselves without concern for others, black or white.”
    —James P. Comer (20th century)

    “In this distribution of functions, the scholar is the delegated intellect. In the right state, he is, Man Thinking. In the degenerate state, when the victim of society, he tends to become a mere thinker, or, still worse, the parrot of other men’s thinking.”
    —Ralph Waldo Emerson (1803–1882)