If is a random variable on . The expected value of is defined as

**Inequalities**

*Jensen’s inequality*. If is convex and

*Holder’s inequality*. If with then

*Cauchy-Schwarz Inequality*: For

Let with .

hence

*Markov’s/Chebyshev’s inequality*. Suppose has , let and let

where

Alternatively suppose and that is -measurable and non-decreasing (note ). Then

Example:

**Integration to the limit**

is uniformly integrable if s.th.

whenever and

**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

Note .

**Computing Expected Values**

*Change of variables formula*. Let with measure , i.e. . If , so that or , then