mirror of
https://github.com/jackyzha0/quartz.git
synced 2025-12-24 21:34:06 -06:00
109 lines
4.0 KiB
Markdown
109 lines
4.0 KiB
Markdown
# 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! |