The binary errors－and－erasures channel is given by （1)Find the capacity （2)Speci

The binary errors-and-erasures channel is given by (1)Find the capacity (2)Specialize to erasures only(ε=0) (3)Specialize to the binary symmetric channel(ρ=0) (4)Would you prefer a binary symmetric channel with crossover probability=0．125 or a simple erasure channel with probability of erasure=0．5？

The binary errors－and－erasures channel is given by

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第1题

Using Venn diagrcuns(维拉图)，we can see that mutual information common to three random var

iable 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 analogy for entropy and mutual lnlormatlon．The second identity follows easily from the first．

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第2题

Conditional mutual information．Consider a sequences of N binary random variables X1，X2，…，X

N．Each sequences an even number of 1’S has probabilitv 2-(N-1) and each sequences with an odd number of 1’s has probability zero．Find the average mutual informations I(X1；X2)，I(X3；X2|X1)，…，I(XN；XN-1|X1X2…XN-2) Check your result for N=3．

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第3题

有一信源输出X∈{0，1，2}，其概率为p0=1／4，p1=1／4，p2=1／2。设计两个独立实验去观察它，其结果分别为Y1

有一信源输出X∈{0，1，2}，其概率为p0=1/4，p1=1/4，p2=1/2。设计两个独立实验去观察它，其结果分别为Y1∈{0，1}和Y2∈{0，1}。已知条件概率为：

求：(1)I(X；Y1)和I(X；Y2)，并判断哪个实验好些。 (2)I(X；Y1Y2)，并计算做Y1和Y2两个实验比做Y1或Y2中的一个实验各可多得多少关于X的信息。 (3)I(X；Y1|Y2)和I(X；Y2|Y1)，并解释它们的含义。