Econ 900

Independence and Identical distribution.

RVs are said to be independent if the outcome of any one Xi does not influence and is not influenced by and other RV Xj.

RVs are identically distributed if their density functions are identical in the sense that f(X1| theta) = f(X2|theta) = f(Xn|theta)

• 1. All possible distinct outcomes are known a-priori
• 2. In any particular trial the outcome is not known a-priori but there exists a discernible regularity of occurrence associated with these outcomes.
• 3. The experiment can be repeated under identical circumstances.

• All possible distinct outcomes of an experiment.
• For flipping a coin S={H,T}
• For SAT scores S={200,210,220,...,780,790,800}
• For reaction time to a stimulus S would consist of all positive numbers i.e. S=(0, OO)

• An event is a collection of possible outcomes of an experiment, that is any subset of S (including S itself).
• Let A be an event, a subset of S. We say the event A occurs if the outcome of the experiment is in A.

A is a subset of S ( A C S or A C S) if every element of A is also in S.

Union of sets A and B ( A U B) is all the elements that are in both or either sets.

Intersection of A and B ( A ∩ B) is all the elements are are in BOTH A and B.

The compliment of set A (A^c) is all the elements that are not a part of A. If A{0,1,2,3,} and S{0,1,2,3,4,5} A^c would be {4,5}

Empty set (Ø) is the impossible event. It does not occur. It contains no elements.

An event space F is a set whose elements are the events of interest as well as the related events i.e. those obtained by combining the events of interest using set theoretic operations. When ∩, U, or the compliment, is applied to any elements of F, the result is also an element of F.

• A non-empty class A of sets is called a field of sets iff it is closed under:
• 1. Finite unions
• 2. Finite intersections
• 3. Compliments

• A collection F of subsets S is said to be a field if it satisfies 3 conditions:
• 1. S ε F
• 2. If A ε F then A^c ε F.
• 3. If A, B ε F then ( A U B) ε F.

• A field is:
• 1. Non-empty due to condition 1
• 2. Closed under complimentation, due to 2
• 3. Closed under finite unions, due to 3.
• 4. Closed under finite intersections due to 2 and 3.

• 1. S ϵ F.
• 2. If A ϵ F, then A^c ϵ F.
• 3. If Ai ϵ F for i=1,2,3...n. the Union of all i's from 1 to infinity ϵ F.

• A non-empty class F of sets is called a sigma field iff it is closed under:
• 1. countable unions
• 2. countable intersections
• 3. compliments

• 1. Ø is in B.
• 2. If A is in B, then Ac is in B (B is closed under complementation)
• 3. If A1, A2, ... is in B, then the Union from i=1 to OO of Ai is in B (B is closed under countable unions)

• B={all subsets of S, including S itself}
• If S={1,2,3} then B is a collection of the following sets:
• {1}{2}{3}{1,2}{1,3}{2,3}{1,2,3}{ Ø }
• Therefore if S has n elements, there are 2ⁿ sets in B

[a,b] (a,b] [a,b) and (a,b) for all real numbers a and b. Also B contains all sets that can be formed by taking (possibly countably infinite) unions and intersections of sets of the above varieties.

A∩B=Ø, or the intersection of A and B has no elements.

Zero, probability for such a variable is measurable only over a given range or interval, such as (a,b).

Discrete or continuous

A discrete RV takes on only finite (or countable infinite) number of values. An example would be the sum of numbers on 2 dice thrown. X can be 2,3,4,5,6,7,8,9,10,11, or 12. I.e. if its range is a countable subset of the R.

A continuous RV is one that can take on any value in some interval. The height of an individual is a continuous RV in the range of say 60-65 inches depending on the precision in the measurement. I.e. if its range of values is any uncountable subset of the R.

Derivative of the (cumulative) distribution function.

• A function that is a mapping from the original sample space S to the real numbers. Or a function that attaches numbers to all elements of S in a way that preserves the event structure of F.
• If the exp is tossing 2 dice, X could be the sum of the numbers.
• If the exp is tossing a coin 25 times, X could be the number of heads in 25 tosses.
• The nature of the RV depends crucially on the size of the field in question. If F is small, being a RV w.rt. F is very restrictive.

(S, F, P( )). The sample space, the field, and the probability set function.

• 1. The set up for the experiment remains the same for all trials.
• 2. The outcome in one trial does not effect that of another. (Independence)

P(A|B)= P(B|A) * P(A) / P(B) for P(B)>0

It describes the probability for each x ϵ S. The density function has the values on the X-axis and their associated probabilities on the Y-axis. By definition the probabilities sum to 1.

• 1. Fx(.) is non-decreasing
• 2. Fx(.) is right continuous
• 3. Lim as x -> OO : = Fx(OO) = 1
• 4. Lim as x -> -OO : = Fx(-OO) = 0

(Cumulative) distribution function

Probability density function

P(A) + P (B)

HH, HT, TH, TT

S=(0, OO)