================================================================================ - Suppose there are 2 random variables. - When samples of one random variable vary, if conditional samples of other random variable vary, you say they're in correlation to each other. - Otherwise, you say they're independent ================================================================================ Independence of random variables $$$f_{XY}(x,y)=f_X(x)f_Y(y)$$$ $$$f_{XY}(x,y)$$$: joint pdf of X and Y $$$f_X(x)$$$: marginal pdf of X $$$f_Y(y)$$$: marginal pdf of Y ================================================================================ Independence of 3 random variables $$$f_{XYZ}(x,y,z)=f_X(x)f_Y(y)f_Z(z)$$$ In that situation, all following are also independent. XY, XZ, YZ, ... ================================================================================ Iterative trial. - Suppose you extract multiple samples from one random variable. - You can consider samples as the samples which are from independent random variables. - Therefore, $$$f(x_1,x_2,\cdots,x_N) \\ =f(x_1)\times f(x_2) \times \cdots f(x_N) \\ =\prod\limits_{i=1}^{N} f(x_i)$$$ f(x): PDF $$$x_1,x_2,\cdots,x_N$$$: sample data $$$f(x_1,x_2,\cdots,x_N)$$$: probability value of $$$x_1,x_2,\cdots,x_N$$$ occuring ================================================================================ Conditional probability distribution - Suppose 2 independent random variables X and Y - Conditional probability distribution of X and Y is identical to marginal PDF $$$f_{X \mid Y} (x | y) = \dfrac{f_{XY}(x, y)}{f_{Y}(y)} = \dfrac{f_{X}(x) f_{Y}(y)}{f_{Y}(y)} = f_{X}(x)$$$ $$$f_{Y \mid X} (y | x) = \dfrac{f_{XY}(x, y)}{f_{X}(x)} = \dfrac{f_{X}(x) f_{Y}(y)}{f_{X}(x)} = f_{Y}(y)$$$ - It means if random variable X is independent to Y, conditional probability distribution is not affected by conditional random variable - That is, $$$f(x|y_1)=f(x|y_2)$$$ ================================================================================ Expectation value of independent random variables - Suppose 2 indenpendent random variable X and Y - Then, following are true $$$E[XY]=E[X]E[Y]$$$ $$$E[(X-\mu_X)(Y-\mu_Y)]=0$$$ ================================================================================ Variance value of independent random variables $$$Var[X+y]=Var[X]+Var[Y]$$$