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Definition
μ=1/n Σ[i=1,n] x_i
σ=√(1/(n-1) Σ[i=1,n] (x_i-x_bar)^2) |
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Definition
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Definition
1) Complement(AnB) =C(A)uC(B)
2) Complement(AuB)=C(A)nC(B) |
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Definition
3) An(BuC)=(AnB)u(AnC)
4) Au(BnC)=(AuB)n(Auc) |
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Definition
1) P(A)≥0
2) P(S)=1
3) P(A1uA2u…uAn) = Σ[i=1,n] P(Ai) |
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Definition
| # of ways of ordering n distinct objects taken r at a time: P[n,r]=n!/(n-r)! |
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Definition
| # of combos of n objects selected r at a time: C[n,r]=(n,r)=n!/(r!(n-r)!) |
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Definition
| The # of ways of partitioning n distinct objects into k distinct groups containing n1,n2,…,nk objects, respectively, where each object appears in exactly one group and Σ[i=1,k] ni=n, is N= (n, n1 n2 … nk)=n!/(n1!n2!...nk!) |
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Definition
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| Multiplicative and Additive Laws |
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Definition
P(AnB)=P(A)P(B|A) [if A,B are independent, =P(A)P(B)]
P(AuB)=P(A)+P(B)-P(AnB) [if A,B are mutually exclusive, =P(A)+P(B)] |
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Definition
| P(A)=Σ[i=1,k] P(A|Bi)P(Bi) |
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Definition
| P(Bj|A) = P(A|Bj)P(Bj)/(Σ[i=1,k] P(A|Bi)P(Bi)) |
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Definition
E(Y)=μ=Σ[y] yP(y)
V(Y)=σ^2=E[(Y-μ)^2]=E[Y^2]-μ^2 |
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Definition
| Moment: a set of numerical measures that describe or uniquely determine P(Y) under certain conditions The kth moment of the RV Y taken around the origin is E(Y^k) and is written as μ'_k |
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Definition
γ_1=(E(Y-μ)^3)/σ^3 (γ can be positive, negative or 0/symmetric)
kurtosis replaces 3 with 4 |
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Definition
1) n identical, fixed trials
2) each trial results in 1 of 2 possible outcomes
3) probability of success is p, failure is 1-p (aka q)
4) trials are independent
5) R.V. Y is the # of successes |
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Definition
P(Y)=(n,y)(p^y)(q^(n-y))
E(Y)=np
V(Y)=npq |
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Definition
1) F(-∞)=0
2) F(∞)=1
3) F(*) is non-decreasing in y ST if y1<y2 then F(y1)≤F(y2) |
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| Rules of non-cooperative games |
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Definition
1) Players ξ={1,2,…,I} where I≥2
2) Moves, feasible actions Ai=[0,∞)
3) Order of moves: sequential or simultaneous
4) Information: perfect (ie perfect recall) or imperfect
5) Payoffs: preferences over outcomes |
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Definition
| A complete contigent play (decision rule) that specifies the player's action at every decision node s/he may encounter. Strategy profile: S={s1,s2,…,sI} |
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Definition
| si€Si is a dominant strategy if u(si,s-i)>ui(si',s-I') A si'€Si, si'≠si and A s-i€S-i |
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Definition
| si€Si is strictly dominated if E si'€Si ST ui(si',s-i)>ui(si,s-i) A s-i€S-i |
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Definition
| There is an NE for the strategy profile S*={s1*,s2*,…,sI*} if for all i=1,…,I ui(si*,s-i*)≥ui(si',s-i*) A si€Si |
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| Ball-Romer payoff function |
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Definition
u_i = w(m/p,p_i/p)-zD_i
FOC: W2(1,1)=0, SOC: W22(1,1)<0, W12>0 implies as income increases firms have an incentive to change prices |
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Definition
d(Pi*/P)/d(M/P) = -W_12/W_22 = π
small W12 can be the efficiency wage and large W22 is an extreme example of monopolistic competition |
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Definition
| PC<z, PC=W(M,Pi*/P) -W(M,1) |
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| Ball-Romer measure of nominal rigidity |
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Definition
the band where price does NOT change is (1-x*,1+x*)
SO Taylor Approximation is PC≈[-(W_12)^2 / 2W_22]x^2
x*=√[(-2zW_22)/(W_12)^2] = √(2z/πW_12) |
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Definition
| Symmetrical Nash Equilibria: S={s€[0,E]|V_1(e,e)=0} where: e_i is an agent's action, e is an action that at the SNE where ei*(e)=e, e-bar is everyone else's action, V(*) is the payoff function, E is the finite bound to an agent's action ei |
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Definition
| Symmetric Cooperative Equilibria: S~ ={e€[0,E]|V1(e,e)+V2(e,e)=0;V11(e,e)+2V12(e,e)+V22(e,e)<0} where if i) V2(ei,e-bar)>0 the game exhibits positive spillovers, ii) if V2(ei,e-bar)<0 the game exhibits negative spillovers, iii) if V12(ei,e-bar)>0 the game exibits strtegic complementarity, iv) if V12(ei,e-bar)<0 the game exhibits strategic substitutability, and v) if dΣej*/dθi>dei*/dθi>δei*/δθi the game exhibits multiplier effects, where θi is a parameter of i's payoff function that we assume to be equal for all i |
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| strictly dominated mixed strategy |
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Definition
mi is S.D. if E mi'€m(si) ST ui(mi',m-i)>ui(mi,m-i) A m-i€M-i
(to replace mixed strategies with a pure strategy, substitute m-I for s-i) |
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| Existence of a Nash Equilibrium |
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Definition
| Proposition: A NE exists if i) si is non-empty, convex and compact in R^n, ii) ui(s1,…,sI) is continuous in (s1,…,sI) and quasi-concave in si |
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Definition
| If f(x) is a function of x and f(x*)=x*, then x* is a fixed point |
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| Brouwer's Fixed Point Theorem |
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Definition
| Let S C R^n be non-empty, convex and compact and f:S->S be continuous, the there exists x*€S ST f(x*)=x* |
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Definition
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| probability density function |
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Definition
| f(y)=dF(y)/dy=F'(y) p(a≤y≤b) = integral[a,b]f(y)dy=F(b)-F(a) |
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| E(Y), E[g(y)] (area under a curve) |
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Definition
| integral [-∞,∞] yf(y)dy, integral [-∞,∞] g(y)f(y)dy |
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| Uniform continuous distribution function, also mean & variance |
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Definition
| f(y) = { 1/(θ2-θ1) for θ1≤y≤θ2; 0 otherwise} E(Y)=(θ1+θ2)/2; V(Y)=σ^2=(θ2-θ1)^2/12 |
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Definition
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Definition
P(y1|y2) = P(y1,y2)/P2(y2), provided p2(y2)>0
f(y1|y2) = f(y1,y2)/f2(y2), provided f2(y2)>0 |
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| Indpendence for discrete and continuous random variables |
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Definition
Discrete: P(y1,y2)=p1(y1)p2(y2)
Continuous: f(y1,y2)=f1(y1)f2(y2) |
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| COV(Y1,Y2), P(Cov. Coefficient) |
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Definition
"COV(Y1,Y2)=E[(Y1-μ1)(Y2-μ2)]=E[Y1,Y2]-μ1μ2, where E[Y1,Y2]=Σ[for all y1]Σ[for all y2] y1y2p(y1,y2),
P(Cov. Coefficient) = COV(Y1,Y2)/σ1σ2
If Y1,Y2 are indpendent then COV(Y1,Y2)=0 (BUT converse is not true)" |
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| Conditional expectation of g(Y1)|Y2=y2 |
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Definition
For jointly continuous Y1, Y2: E(g(Y1)|Y2=y2) = integral [-∞,∞] g(y1)f(y1|y2)dy1;
For jointly discrete Y1, Y2: E(g(Y1|Y2=y2)=Σ[all y1] g(y1)p(y1|y2) |
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Definition
| A function of the observable random variables in a sample and known constants |
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Definition
| indentically and independently distributed |
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Definition
| Central Limit Theorem: Let y1,…,yn be random samples from any distribution with a pop. mean of μ, and variance σ^2, then E(y-bar)=μ, V(y-bar)=σ^2/n and y-bar will have approximately a normal distribution as long as the sample size is iid and sufficiently large, aka y-bar~N(μ,σ^2/n) |
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