Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>1.</strong>4. PROBABILITY MEASURES<br />
(i) P(∅) = 0;<br />
(ii) Let A 1 ,A 2 ,...,A n ∈ A with A i ∩A j = ∅, i ≠ j. Then,<br />
P<br />
(<br />
⋃ n<br />
)<br />
A i =<br />
i=1<br />
n∑<br />
P(A i ).<br />
i=1<br />
(iii) ∀A ∈ A, P(A) ≤ <strong>1.</strong><br />
(iv) ∀A ∈ A, P(A c ) = 1−P(A).<br />
(v) For A,B ∈ A with A ⊆ B, it holds P(A) ≤ P(B).<br />
(vi) Let A,B ∈ A be two events. Then<br />
P(A∪B) = P(A)+P(B)−P(A∩B).<br />
(vii) Let A 1 ,A 2 ,...,A n ∈ A be events. Then<br />
( n<br />
)<br />
⋃ n∑ n∑<br />
P A i = P(A i )− P(A i1 ∩A i2 )<br />
i=1 i=1 i 1 ,i 2 =1<br />
i 1