If are pairwise independent and then as

Proof:

Let and . Since are pairwise independent, the are uncorrelated and thus

Since

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# Category: Probability

# Almost sure convergence via pairwise independence

# Karl Popper: Conjectures and Refutations

# Expectation: Useful properties and inequalities

# Very brief notes on measures: From σ-fields to Carathéodory’s Theorem

If are pairwise independent and then as

Proof:

Let and . Since are pairwise independent, the are uncorrelated and thus

Since

Continue reading “Almost sure convergence via pairwise independence”

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(1) It is

easy to obtain confirmations, or verifications, for nearly every theory-if we look for confirmations.(2) Confirmations

should count only if they are the result of risky predictions; that is to say, if, unenlightened by the theory in question, we should have expected an event which was incompatible with the theory–an event which would have refuted the theory.(3)

Every ‘good’ scientific theory is a prohibition: it forbids certain things to happen. The more a theory forbids, the better it is.(4)

A theory which is not refutable by any conceivable event is nonscientific. Irrefutability is not a virtue of a theory (as people often think) but a vice.(5)

Every genuine test of a theory is an attempt to falsify it, or to refute it. Testability is falsifiability; but there are degrees of testability: some theories are more testable, more exposed to refutation, than others; they take, as it were, greater risks.(6) Confirming evidence should not count except when it is the result of a

genuine testof the theory; and this means that it can be presented as a serious but unsuccessful attempt to falsify the theory. (I now speak in such cases of ‘corroborating evidence’.)(7) Some genuinely testable theories, when found to be false, are still upheld by their admirers–for example by introducing ad hoc some auxiliary assumption, or by re-interpreting theory ad hoc in such a way that it escapes refutation. Such a procedure is always possible, it rescues the theory from refutation only

at the price of destroying, or at least lowering, scientific status.

—————————-

Excerpt from a lecture given by Karl Popper at Peterhouse, Cambridge, in Summer 1953, as part of a course on Developments and trends in contemporary British philosophy.

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

Continue reading “Expectation: Useful properties and inequalities”

**Definition 1.** A -field is a non-empty collection of subsets of the sample space closed under the formation of complements and countable unions (or equivalently of countable intesections – note ). Hence is a -field if

whenever

whenever

**Definition 2. Set functions and measures**. Let be a set and be an algebra on , and let be a non-negative set function

- is
**additive**if and, for , - The map is called
**countably additive**(or -additive) if and whenever is a sequence of disjoint sets in with union in , then - Let be a measurable space, so that is a -algebra on .
- A map is called a
**measure**on if is countable additive. The triple is called a**measure space**. -
The measure is called

**finite**ifand –

**finite**if, () s.th.

- Measure is called a
**probability measure**if and is then called a**probability triple**. - An element of is called -null if .
- A statement about points of is said to hold
**almost everywhere**(a.s.) if

** Continue reading “Very brief notes on measures: From σ-fields to Carathéodory’s Theorem” **