(New page: Category:ECE600 Category:Math == Theorem == Let A<math>_1</math>,A<math>_2</math>,...,A<math>_n</math> be n pairwise disjoint events in event space. Then <br/> <center><math>P(\...) |
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==Proof == | ==Proof == | ||
− | By Kolmogorov axioms, for pairwise disjoint events A<math>_1</math> and A<math>_2</math> in ''F'', <br/> | + | By Kolmogorov axioms, for pairwise disjoint events A<math>_1</math> and A<math>_2</math> in event space ''F'', <br/> |
<center><math>P(A_1\cup A_2) = P(A_1)+P(A_2)</math></center> | <center><math>P(A_1\cup A_2) = P(A_1)+P(A_2)</math></center> | ||
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Let <br/> | Let <br/> | ||
<center><math>B = \bigcup_{i=1}^n A_i</math></center> | <center><math>B = \bigcup_{i=1}^n A_i</math></center> | ||
− | Note that B ∩ A<math>_{n+1}</math> = ø because if x ∈ A<math>_{n+1}</math>, then x ∉ A<math>_i</math> ∀i = 1,2,...,n ⇒ x ∉ B. On the other hand if x ∈ B, then x ∈ A<math>_i</math> for exactly | + | Note that B ∩ A<math>_{n+1}</math> = ø because if x ∈ A<math>_{n+1}</math>, then x ∉ A<math>_i</math> ∀i = 1,2,...,n ⇒ x ∉ B. On the other hand if x ∈ B, then x ∈ A<math>_i</math> for exactly one i since A<math>_i</math>'s are pairwise disjoint. A<math>_{n+1}</math> and A<math>_i</math> are also disjoint so x ∉ A<math>_{n+1}</math> |
We mentioned earlier that the probability of the union of two disjoint events is the sum of the probabilities of those events. Hence,<br/> | We mentioned earlier that the probability of the union of two disjoint events is the sum of the probabilities of those events. Hence,<br/> | ||
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− | Thus by induction, we have that for | + | Thus by induction, we have that for a finite collection of pairwise disjoint events A<math>_1</math> through A<math>_n</math> in ''F'', <br/> |
<center><math>P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^n P(A_i)</math></center> | <center><math>P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^n P(A_i)</math></center> | ||
<math>\blacksquare</math> | <math>\blacksquare</math> |
Latest revision as of 14:47, 15 October 2013
Theorem
Let A$ _1 $,A$ _2 $,...,A$ _n $ be n pairwise disjoint events in event space. Then
Proof
By Kolmogorov axioms, for pairwise disjoint events A$ _1 $ and A$ _2 $ in event space F,
Induction Hypothesis: Let A$ _1 $,A$ _2 $,...,A$ _n $ be pairwise disjoint events in F, i.e.
Then,
Now, I want to show that for pairwise disjoint events A$ _i $ i = 1,2,...,n+1
Let A$ _{n+1} $ ∈ F such that A$ _{n+1} $ is pairwise disjoint from events A$ _i $, i = 1,2,...,n.
Let
Note that B ∩ A$ _{n+1} $ = ø because if x ∈ A$ _{n+1} $, then x ∉ A$ _i $ ∀i = 1,2,...,n ⇒ x ∉ B. On the other hand if x ∈ B, then x ∈ A$ _i $ for exactly one i since A$ _i $'s are pairwise disjoint. A$ _{n+1} $ and A$ _i $ are also disjoint so x ∉ A$ _{n+1} $
We mentioned earlier that the probability of the union of two disjoint events is the sum of the probabilities of those events. Hence,
Thus by induction, we have that for a finite collection of pairwise disjoint events A$ _1 $ through A$ _n $ in F,
$ \blacksquare $