Econ 900

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Econ 900
2010-09-26 15:11:37
Graduate econometrics probability statistics

material for the first test
Show Answers:

  1. What does the randomness assumption imply?
    Independence and Identical distribution.
  2. What is statistical independence?
    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.
  3. What is identical distribution?
    RVs are identically distributed if their density functions are identical in the sense that f(X1| theta) = f(X2|theta) = f(Xn|theta)
  4. What are the 3 conditions for a random experiment
    • 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.
  5. Define Sample Space
    • 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)
  6. Define Event
    • 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.
  7. Subset
    A is a subset of S ( A C S or A C S) if every element of A is also in S.
  8. Union
    Union of sets A and B ( A U B) is all the elements that are in both or either sets.
  9. Intersection
    Intersection of A and B ( A ∩ B) is all the elements are are in BOTH A and B.
  10. Compliment
    The compliment of set A (Ac) 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} Ac would be {4,5}
  11. Empty Set
    Empty set (Ø) is the impossible event. It does not occur. It contains no elements.
  12. Define Event Space
    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.
  13. What is the definition of a field?
    • 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
  14. Define the 3 conditions for a set to be a field
    • 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 Ac ε F.
    • 3. If A, B ε F then ( A U B) ε F.
  15. What are the four characteristics of a field?
    • 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.
  16. What are the 3 conditions for a field to be a sigma (δ) or sigma algebra field?
    • 1. S ϵ F.
    • 2. If A ϵ F, then Ac 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.
  17. What is the definition of a sigma field?
    • 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
  18. Define (B) the borel field's 3 properties.
    • 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)
  19. If S is finite or countable, then for a given sample space S the borel field is ...
    • 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
  20. Let S=(-OO,OO), the real line, then the borel field is ...
    [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.
  21. Mathematically describe what two mutually exclusive (or disjoint) events A and B look like
    A∩B=Ø, or the intersection of A and B has no elements.
  22. For a continuous random variable, the probability that it takes a specific value is ?
    Zero, probability for such a variable is measurable only over a given range or interval, such as (a,b).
  23. A random variable may be either ?
    Discrete or continuous
  24. Define a discrete RV
    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.
  25. Define a continuous RV
    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.
  26. The probability density function of a continuous variable is defined as the ...
    Derivative of the (cumulative) distribution function.
  27. What is a random variable?
    • 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.
  28. What are the 3 things that make up a probability set space?
    (S, F, P( )). The sample space, the field, and the probability set function.
  29. What 2 things are needed for a random trial?
    • 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)
  30. What is Bayes' Formula?
    P(A|B)= P(B|A) * P(A) / P(B) for P(B)>0
  31. What is a density function?
    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.
  32. What are the 4 properties of the (cumulative) distribution function?
    • 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
  33. The integral of the PDF is the ?
    (Cumulative) distribution function
  34. The derivative of the (C) DF is the ?
    Probability density function
  35. Axiom of Finite additivity: If A ε B" and B ε B" are disjoint (mutually exclusive), then P( A U B) =
    P(A) + P (B)
  36. What is the sample space of : tossing a coin 2 times?
    HH, HT, TH, TT
  37. What is the sample space of the lifetime in hours of a particular brand of light bulb?
    S=(0, OO)