Stochastic

# 13 in the discrete case a13 is equivalent to p x x

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Unformatted text preview: P (A) = P (B ) = P (C ) = 1/2. The intersection of any two of these events is A \ B = B \ C = C \ A = {HHH, T T T } an event of probability 1/4. From this it follows that P (A \ B ) = 11 1 = · = P (A)P (B ) 4 22 i.e., A and B are independent. Similarly, B and C are independent and C and A are independent; so A, B , and C are pairwise independent. The three events A, B , and C are not independent, however, since A \ B \ C = {HHH, T T T } and hence ✓ ◆3 1 1 P (A \ B \ C ) = 6= = P (A)P (B )P (C ) 4 2 The last example is somewhat unusual. However, the moral of the story is that to show several events are independent, you have to check more than just that each pair is independent. A.2. RANDOM VARIABLES, DISTRIBUTIONS A.2 211 Random Variables, Distributions Formally, a random variable is a real-valued function deﬁned on the sample space. However, in most cases the sample space is usually not visible, so we describe the random variables by giving their distributions. In the discrete case where the random variable can take on...
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