quartz/content/vault/econometrics/lectnot/Exam 1 Review.md

Exam 1 Review

1. Spiritual thought: prayer through life's trials, faith without works is dead, obedience gives confidence

2. Exam info

Exam 1 Review

1. Spiritual thought: prayer through life's trials, faith without works is dead, obedience gives confidence

2. Exam info

   • Any calculator

   • No time limit, predict 2-3 hours

   • Handout provided

   • Hard: typically C average

3. Study tips

   • Take it seriously: equal weight with final exam

   • Start with study guide

   • Practice exams (first without solutions, then with)

   • Extra homework problems (or repeat homework problems)

   • Time saver: talk through steps, don't work out algebra

4. Exam strategies

   • Don't forget to pray!

   • Extend familiar material to unfamiliar settings (good practice for real world)

   • Difficulty is similar to homework, but no TAs or books, so fewer A's than homework

   • Average score is typically C, which averaged with A- homework gives B- overall.

   • Show work and list what you know for partial credit (e.g. \rho =\frac{\sigma_{\text{xy}}}{\sigma_{x}\sigma_{y}}, even if you can't figure out \sigma_{\text{xy}})

5. Key formulas

   • Binary events

     1. P(E) =\frac{\# E}{\# S}
     2. C_{k}^{n} =\frac{n!}{k!(n - k)!}
     3. P(A\cup B) = P(A) + P(B) - P(A\cap B)
     4. Independent events: P(A\cap B) = P(A)P(B) (or P(B) = P(A))
     5. P(B) =\frac{P(A\cap B)}{P(B)}
     6. P(A\cap B) = P(B|A)P(A)

   • Random variables

     1. Legitimate distribution? \sum P(x) =\int f(x)dx = 1
     2. Mode maximizes P(x) or f(x) (i.e. f^{'}(x) = 0 and f^{''}(x) < 0)
     3. \mu = E(X) =\sum xP(x) =\int xf(x)\text{dx}
     4. E(X^{3}) =\sum x^{3}P(x) =\int x^{3}f(x)\text{dx}
     5. \sigma^{2} = V(X) = E\lbrack(X -\mu)^{2}\rbrack = E(X^{2}) -\mu^{2}; \sigma =\sqrt{V(X)}
     6. F(x) = \int_{-\infty}^xf(\widetilde{x})d\widetilde{x}, f(x) = F'(x)
     7. P(a < X < b) = F(b) - F(a)
     8. Percentile: solve F(\phi_{.5}) = .5
     9. f(x) = F'(x)

   • Joint distributions

     1. Legitimate joint distribution? \sum\sum P(x,y) =\iint_{}^{}f(x,y)dxdy = 1
     2. Marginal distribution P_{x}(x) = P(x,y) f_{x}(x) =\int f(x,y)\text{dy}
     3. Independent variables P(x,y) = P_{x}(x)P_{y}(y) f(x,y) = f_{x}(x)f_{y}(y)
     4. E(\frac{X}{Y}) =\sum\sum(\frac{x}{y})P(x,y) =\iint_{}^{}(\frac{x}{y})f(x,y)\text{dxdy}
     5. \text{Cov}(X,Y) = E\lbrack(X -\mu_{x})(Y -\mu_{y})\rbrack = E(\text{XY}) -\mu_{x}\mu_{y}
     6. \rho =\frac{\text{Cov}(X,Y)}{\sigma_{x}\sigma_{y}}
     7. Conditional distribution P(X=x|Y = 3) =\frac{P(x,3)}{P_{y}(3)} f_{x}(x|Y = 3) =\frac{f(x,3)}{f_{y}(3)}
     8. E(X|Y = 3) =\sum xP(x|Y = 3) =\int xf(x|Y = 3)\text{dx}
     9. V(X|Y = 3) = E(X^2|Y = 3) -\lbrack E(X|Y = 3)\rbrack^{2}

   • Regressions

     1. \beta_{1} =\frac{\sigma_{\text{xy}}}{\sigma_{x}^{2}} =\rho\frac{\sigma_{y}}{\sigma_{x}}
     2. \beta_{0} =\mu_{y} - b\mu_{x}
     3. \frac{V(a + bX)}{V(Y)} =\rho^{2}
     4. \varepsilon_{i} = Y_{i} -(\beta_{0} +\beta_{1}X_{i})

6. Algebra tricks

   • E(\$ 100 -\$ 5X) =\$ 100 -\$ 5E(X)

   • V(\$ 100 -\$ 5X +\$ 3Y) = V(\$ 100) + V(-\$ 5X) + V(\$ 3Y) + 2Cov(-\$ 5X,\$ 3Y) = 0 +(\$ 5)^{2}V(X) +(\$ 3)^{2}V(Y) -\$ 30Cov(X,Y)

   • \text{Cov}(\$ 100 -\$ 5X,Y) = Cov(\$ 100,Y) + Cov(-\$ 5,Y) = 0 -\$ 5Cov(X,Y)

   • \text{Corr}(\$ 100 -\$ 5X,Y) = Corr(- X,Y) = - Corr(X,Y)

7. Distributional relationships

   • If X\sim N then 3X\sim N and X + 7\sim N

   • If X_{1},X_{2}\sim N then X_{1} + X_{2}\sim N

   • If Z\sim N(0,1) then Z^{2}\sim\chi^{2}(1)

   • If W_{1}\sim\chi^{2}(3),W_{2}\sim\chi^{2}(5) independent then W_{1} + W_{2}\sim\chi^{2}(8) and \frac{W_{1}/3}{W2/5}\sim F(3,5)

   • If Z\sim N(0,1) and W\sim\chi^{2}(\nu) independent then \frac{Z}{}\sim t(\nu)

8. Rejoice in how much we've learned!