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By: Neil E. Cotter |
Probability |
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Independent discrete RV |
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Example 1 |
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Ex: The following formulas define the behavior of conditional probabilities:
(always true)
(if A and B independent)
(if A and B independent)
For the following formulas, determine whether the formula is always true when A and B are independent.
a)
b)
c) If and , then
d) For an arbitrary event, C, A is independent of .
Sol'n: a) The first equation follows by direct application of the formulas for independent events:
b) Because A and B are independent, we may immediately simplify the right-hand side of the equation:
Now we consider the left side of the equation:
Using the Law to Total Probability, we may relate to :
or
Again using the Law of Total Probability, we may relate to .
or
Substituting into our equation for , we have the following result:
or
The left side of the original equation now simplifies to :
Thus, the left and right sides of the original equations are equal whenever A and B are independent:
Note: Our derivation shows that, when A and B are independent events, we also have independence of A and B ', A' and B, and A' and B '.
c) If and , then follows immediately. What is less immediately obvious is that this result implies that . In other words, the intersection of A and B is nonempty. Equivalently, A and B must overlap on a Venn diagram.
d) For an arbitrary event, C, we investigate the independence of A and by examining the conditional probability for A.
Since may be any part of B, we may consider the case where :
Since it is not always true that P(A) = 1, the equation in (d) is not always true.