A source has five letters with following probabili

A source has five letters with following probabilities：P(a1)=0．3，P(a2)=0．2，P(a3)=0．2，P(a4)=0．15，P(a5)=0．15 These letters are to be coded into binary digits for use on a noiseless channal．It takes 1 second to transmit a 0 and 3 seconds to transmit a 1．Using cut and try techniques，find a code with the prefix condition that minimizes the average time required to transmit a source letter and calculate this minimum average time． (2)Any such code can be represented by a tree in which the length of a branch：is proportional to the time required to transmit the associated digit．Show that for a code to minimize the average transmission time，the probabilities associated with intermediate and terminal nodes must be nonincreasing with length．
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