二つの変数によるガウス確率分布の続き
In fact, it is easier to solve for the general multivariate Gaussian and then substitute for x with n=2. For x as am n-dimensional vector of independent normal random variables with mean 0 and variance 1, f(x) = (1/(2pi))^(n/2)e^*1.
By the formula for changing variables in a pdf, f(y;p) is derived by solving for x in terms of y and computing the Jacobian of x. Then n can be set to 2, sigma expressions rewritten in terms of the correlation p (after simplifying matrix products and determinants involving the square root of the covariance matrix), and a 2-dimensional y substituted. The final pdf is f(y1,y2;p)=(1/(2pi))(1/(sigma1sigma2*sqrt(1-p^2)))exp*2+((y2-mu2)/sigma2^2)))
Note that 1/(1-p)^2 =(sigma1^2*sigma2^2)/(sigma11sigma22-sigma12sigma21)
The square root of the covariance matrix is
| sigma11 sigma12 |
| sigma21 sigma22 |
最近一つの変数のための正規分布を基にして二つの変数による正規分布の計算式を導きたい。しかし簡単に二つの一変数方程式を組み合わせるものではない。
The key to determining the bivariate normal distribution's density function for two correlated normally distributed variables y1 and y2 is to define them in terms of two independent standard normal variables x1 and x2. The density function for these two is obtained trivially by taking the product of one-variable Gaussians:
f(x1,x2)dx1dx2 = (1/(2pi))exp*3
The key to solving for f(y1,y2;p) is to find (x1^2+x2^2) and dx1dx2 in terms of y1,y2, dy1, and dy2.
