Using Venn diagrcuns(维拉图)，we can see that mutual information common to three random variable X，Y and Z should be defined by I(X；Y；Z)=I(X；Y) - I(X；Y|Z) This quantity is symmetric in X，Y and Z，despite the preceding asymmetric define．Unfortunately，I(X；Y；Z)is not nesessarily nonnegative．Find X，Y，Z such that I(X；Y；Z)<0,and Prove the following two identities： I(X；Y；Z)=H(XYZ) - H(X) - H(Y) - H(Z)+I(X；Y)+I(Y；Z)+I(Z；X) I(X；Y；Z)=H(XYZ) - H(XY) - H(YZ) - H(ZX)+H(X)+H(Y)+H(Z) The first identity can be understood using the Venn diargram ogy for entropy and mutual lnlormatlon．The second identity follows easily from the first．