If is a random variable on . The expected value of is defined as
- Jensen’s inequality. If is convex and
- Holder’s inequality. If with then
- Cauchy-Schwarz Inequality: For
Let with .
- Markov’s/Chebyshev’s inequality. Suppose has , let and let
Alternatively suppose and that is -measurable and non-decreasing (note ). Then
Integration to the limit
is uniformly integrable if s.th.
Monotone Convergence Theorem: If then .
Lemma: A measurable function is integrable iff implies
Fatou’s lemma. If then
Dominated Convergence Theorem. If , , and , then
Note when is constant Bounded Convergence Theorem.
Theorem: Suppose and continous functions with i) and when large. ii) as , and iii) . Then
Sums of non-negative Random Variables: Collection of useful results
- If and then
- If , then (see Monotone Convergence Theorem)
- If s.th. , then
- First Borel-Cantelli Lemma. Suppose sequence of events s.th. . Set . Then
and by previous result
Computing Expected Values
Change of variables formula. Let with measure , i.e. . If , so that or , then