Mathematical Methods in the Physical Sciences · Problem 16.3.7

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · probability-theorems-independence

Problem

(a) Note that (3.4) assumes P(A)≠0P(A) \neq 0 since PA(B)P_A(B) is meaningless if P(A)=0P(A) = 0. Assuming both P(A)≠0P(A) \neq 0 and P(B)≠0P(B) \neq 0, show that if (3.4) is true, then P(A)=PB(A)P(A) = P_B(A); that is, if BB is independent of AA, then AA is independent of BB. If either P(A)P(A) or P(B)P(B) is zero, then we use (3.5) to define independence. (b) When is an event EE independent of itself? When is EE independent of ``not EE''?

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