The two-dimensional normal distribution in probability theory. The book says that even if both X and Y obey normal distribution, even if the correlation coefficient is equal to zero, the joint distrib

The two-dimensional normal distribution in probability theory. The book says that even if both X and Y obey normal distribution, even if the correlation coefficient is equal to zero, the joint distribution of xy may not be established. Independence means that a correlation coefficient of 0 is not necessarily independent.

P=0 and xy are independent and necessary and sufficient conditions for each other. If xy obeys two-dimensional normal distribution, xy is normal distribution respectively. P=0 cannot infer that xy is independent, so it cannot be inferred that XY follows a two-dimensional normal distribution.

As long as it is a 2-dimensional normal, then two edges obey the 1 dimensional normal, and the linear combination of any two RVs also obeys the 1 dimensional normal. It doesn't matter whether the two RVs are independent, that is, the correlation coefficient is not equal to 0.

Graphic feature

Concentration: The peak of the normal curve is located in the center, that is, where the average value is located.

Symmetry: The normal curve is centered on the mean value, which is symmetrical left and right, and both ends of the curve never intersect with the horizontal axis.

Uniform variation: the normal curve starts from the place where the mean value is located and gradually decreases evenly to the left and right sides respectively.

The area between the curve and the horizontal axis is always equal to 1, and the probability of the function equivalent to the probability density function integrating from positive infinity to negative infinity is 1. That is, the sum of frequencies is 100%.