Math Concepts

Question 1

[2 Brainteasers] Σ_n=1..N n² = ?

Answer 1

Σ_n=1..N n²​ = N(N+1)(2N+1)/6 = N³/3 + N²/2 + N/6

Question 2

[3-1 Calc] a^x = e^? d a^x/dx = ?

Answer 2

a^x = e^{x lna}

d a^x/dx = a^xlna

Question 3

[3-1 Calc] d(lnx)/dx = ?

Answer 3

d(lnx)/dx = 1/x

Question 4

[3-1 Calc] e

e^x = lim_? (1+?)^?

Taylor series f(x) = e^x = ?

Answer 4

e^x = lim_n→inf (1+x/n)ⁿ

e^x = Σ_n=0..inf [1/n! xⁿ]

Question 5

[3-1 Calc] Taylor series: f(x) = ?

Answer 5

f(x)

= f(x₀) + f’(x₀)(x-x₀) + 1/2 f’‘(x₀)(x-x₀)² + …

= Σ_n=0..inf [1/n! f⁽ⁿ⁾(x₀) (x-x₀)ⁿ]

Question 6

[3-1 Calc] Newton’s method vs. Secant method

Answer 6

Newton’s method: x[t+1] = x[t] - f(x[t]) / f’(x[t]), quadratic convergence

Secant method: x[t+1] = x[t] - f(x[t]) / (f(x[t])-f(x[t-1]))/(x[t]-x[t-1]), (1+5^0.5)/2 convergence

Question 7

[3-1 Calc] Gradient descent

Answer 7

Gradient descent: x[t+1] = x[t] - l * g’(x[t]), l = step size, g’ = grad(g)

(Newton’s method equivalent: f = g’; requires 2nd derivative g’’)

Question 8

[3-1 Calc] Lagrange multipliers

min f(x1,x2) s.t. g1(x1,x2)=0, g2(x1,x2)=0

solve x1, x2

Answer 8

L = f(x1,x2) - l1*g1(x1,x2) + l2*g2(x1,x2)

grad(L) = 0:

dL/dx1 = df/dx1 + l1*dg1/dx1 + l2*dg2/dx1 = 0

dL/dx2 = df/dx2 + l1*dg1/dx2 + l2*dg2/dx2 = 0

dL/dl1 = g1 = 0

dL/dl2 = g2 = 0

Question 9

[3-2 LinAl] proj_v(u) = ?

Answer 9

proj_v(u) = |u| cos(o) v/|v| = |u| u.v/|u||v| v/|v| = u u.v/|v|^2

Question 10

[3-2 LinAl] eigenvalues (v)

Π v = ?

Σ v = ?

det(A - vI) = ?

Answer 10

Π v = det(A)

Σ v = trace(A) = Σ A[i,i]

det(A - vI) = 0

Question 11

[3-2 LinAl] determinants

det(A’) ? det(A’)

det(A^-1) ? det(A)^-1

det(AB) ? det(A)det(B)

A=LR: det(A) ? det(L)det(R) ? (ΠL[i,i]) (ΠR[i,i])

Answer 11

det(A’) = det(A’)

det(A^-1) = det(A)^-1

det(AB) = det(A)det(B)

A=LR: det(A) = det(L)det(R) = (ΠL[i,i]) (ΠR[i,i])

Question 12

[3-2 LinAl] pos semi def

x’Ax ? 0

eigval ? 0

det(A_k) ? 0

Answer 12

x’Ax >= 0

eigval >= 0

det(A_k) >= 0 for all k

Question 13

[3-2 LinAl] matrix derivatives

d/dx a’x = ?

d/dx Ax = ?

d/dx x’Ax = ?

d²/dxdx’ x’Ax = ?

d/dx (Ax+b)’C(Dx+e) = ?

Answer 13

d/dx a’x = a

d/dx Ax = A

d/dx x’Ax = (A’+A)x

d²/dxdx’ x’Ax = A

d/dx (Ax+b)’C(Dx+e) = A’C(Dx+e) + D’C’(Ax+b)

Question 14

[3-2 LinAl] simulate X ~ N(U,S)

Answer 14

X = U + R’Z, where S = R’R and Z~N(0,1)

Question 15

[4 Prob] geometric series

Σ_k=0..n [ar^k] = ?

Answer 15

Σ_k=0..n [ar^k] = a (1-rⁿ⁺¹)/(1-r)

Question 16

[4 Prob] binomial theorem

(a+b)ⁿ = Σ?

Answer 16

(a+b)ⁿ = Σ_k=0..n [nCk a^n-k b^k]

Question 17

[4 Prob] Distribution discrete/continuous: Uniform distribution Unif(a,b)

pmf P(X=x) = ?

pdf f(x) = ?

E[X] = ?

Var(X) = ?

Answer 17

pmf P(X=x) = 1/(b-a+1)

pdf f(x) = 1/(b-a)

E[X] = (b+a)/2

Var(X) = (b-a)²/12 cont, ((b-a+1)²-1)/12 disc

Question 18

[4 Prob] Distribution discrete: Bernoulli Bernoulli(p)

pmf P(X=x) = ?

E[X] = ?

Var(X) = ?

Answer 18

Bernoulli: # success in 1 trial

pmf P(X=x) = p^x(1-p)^1-x, x = 0,1

E[X] = p

Var(X) = p(1-p)

Question 19

[4 Prob] Distribution discrete: Binomial Bin(n,p)

pmf P(X=x) = ?

E[X] = ?

Var(X) = ?

Answer 19

Binomial: # success in n trials

P(X=x) = nCx p^x (1-p)^n-x

E[X] = np

Var(X) = np(1-p)

Question 20

[4 Prob] Distribution discrete: MultiNomial MultiNomial(p1,..pk)

pmf P(X1=x1,..,Xk=xk) = ?

E[Xi] = ?

Var(Xi) = ?

Answer 20

MultiNomial: a specific combo

P(X1=x1,..,Xk=xk) = (n!/x1!..xk!) p1^x1..pk^xk, n = Σxi

E[Xi] = npi

Var(Xi) = npi(1-pi)

Question 21

[4 Prob] Distribution discrete: Hyper Geometric HypGeo(n,N,k)

pmf P(X=x) = ?

E[X] = ?

HypGeo_? –> Bin(n, p=k/N)

Answer 21

Hyper Geometric: # success in n trials w/o replacement

P(X=x) = kCx (N-k)C(n-x) / NCn

E[X] = nk/N

HypGeon<5%N –> Bin(n, p=k/N)

Question 22

[4 Prob] Distribution discrete: Multivariate Hyper Geometric MultiHypGeo(k1,..kk)

pmf P(X1=x1,..Xk=xk) = ?

Answer 22

Multivariate Hyper Geometric: a specific combo w/o replacement

pmf P(X1=x1,..Xk=xk) = (k1Cx1)..(kkCxk)/NCn, N = Σki, n = Σxi

Question 23

[4 Prob] Distribution discrete: Geometric Geo(p)

pmf P(X=x) = ?

cdf F(x) = ?

E[X] = ?

Var(X) = ?

Answer 23

Geometric: # tries to get 1st success

P(X=x) = (1-p)^x-1 p

F(x) = 1-(1-p)^x

E[X] = 1/p

Var(X) = (1-p)/p²

Question 24

[4 Prob] Distribution discrete: Negative Binomial NegBin(r, p)

pmf P(X=x) = ?

E[X] = ?

Var(X) = ?

Answer 24

Negative Binomial: # tries to get rth success

P(X=x) = (x-1)C(r-1) (1-p)^x-r p^r

E[X] = r/p

Var(X) = r(1-p)/p²

Question 25

[4 Prob] Distribution discrete: Poisson Pois(l) pmf P(N(t) = x) = ? E[N(t)] = ? Var(N(t)) = ?

Answer 25

Poisson: # occurrences over an interval P(N(t) = x) = e^-lt (lt)^x / x! E[N(t)] = Var(N(t)) = lt

Question 26

[4 Prob] Distribution continuous: Chi-Sq X²_k, t T_k, F F_k1,k2 X²_k = f(Z), f=? T_k = f(Z, X²_k), f=? F_k1,k2 = f(X²_k1, X²_k2), f=?

Answer 26

X²_k = Σ_i=1..k Z_i² T_k = Z / sqrt(X²_k/k); lim_k→inf = Z F_k1,k2 = (X²_k1/k1) / (X²_k2/k2)

Question 27

[4 Prob] Distribution continuous: Normal univariate N(u,s) and multivariate N(U,S) pdf f(x) = ?

Answer 27

f(x) = 1/sqrt(2πs²) \* e^(-1/2 ((x-u)/s)²) f(x) = 1/sqrt((2π)^k det(S)) \* e^(-1/2 (x-U)'(S^-1)(x-U))

Question 28

[4 Prob] Distribution continuous: Exponential Exp(l) pdf f(x) = ? cdf F(x) = ? E[X] = ? Var(X) = ?

Answer 28

Exponential: time until event occurs (memoryless) f(x) = le^-lx if l \>= 0 else 0 F(x) = 1 - e^-lx E[X] = 1/l Var(X) = 1/l²

Question 29

[4 Prob] MGF M(t) = ? and how does it generate moments?

Answer 29

M(t) = E[e^tX] M'(t) = E[Xe^tX] --\> M'(0) = E[X] M''(t) = E[X²e^tX] --\> M''(0) = E[X²] M⁽ⁿ⁾(t) = E[Xⁿe^tX] --\> M⁽ⁿ⁾(0) = E[Xⁿ]

Question 30

[5 Stochastics] Markov Chain x[i] = prob of reaching s from i x[s] = ? x[i] = ? for all i absorb and not s x[i] = ? for all i transient y[i] = time to absorption y[i] = ? for all i absorb y[i] = ? for all i transient

Answer 30

x[i] = prob of reaching s from i x[s] = 1 x[i] = 0 for all i absorb and not s x[i] = Σ p[i,j]x[j] for all i transient y[i] = time to absorption y[i] = 0 for all i absorb y[i] = Σ p[i,j]y[j] for all i transient p[i,j] = p(i--\>j)

Question 31

[5 Stochastics] Martingale E[Sn] = ? E[Sn²] = ?

Answer 31

E[Sn] = 0, E[Sn²] = n

Question 32

[Stats] ZScores Within 1 std = ?%, 2 std = ?%, 3 std = ?% One-sided 1% = ? std, 5% = ? std, 10% = ? std Two sided 1% = ? std, 5% = ? std, 10% = ? std

Answer 32

Within 1 std = 68.2%, 2 std = 95.4%, 3 std = 99.7% One-sided 1% = 2.33 std, 5% = 1.64 std, 10% = 1.28 std Two sided 1% = 2.58 std, 5% = 1.96 std, 10% = 1.64 std

Question 33

[Stats] Test for Diff H₀: mean(X1) - mean(X2) = d test = ?

Answer 33

if Var(X1) = Var(X2): Z = (mu1 - mu2 - d) / s_p sqrt(1/n1 + 1/n2); s²_p = s1²(n1-1) + s2²(n2-1)/(n1+n2-2) if Var(X1) != Var(X2): Z = (mu1 - mu2 - d) / sqrt(s1²/n1 + s2²/n2)

Question 34

[Stats] 1-way ANOVA H₀: ? test = ?

Answer 34

H₀: u₁=u₂=..=u_K N obs, K groups: SumSqTotal = Σ_k,j (X_k,j - u_X)² = GroupSumSq + WithinGroupSumSq SumSqAcross = Σ_k=1..K n_k(u_k - u_X)² SumSqWithin = Σ_k=1..K (Σ_j=1..J(k) (X_i,j - u_k)²) = Σ_k=1..K (n_k-1)s²_k F = SSA/(k-1) / SSW/(n-k) F ~ 1: H₀ (SSA/(k-1) and SSW/(n-k) are independent estimators of SST/(n-1)) F \>\> 1: not H₀

Question 35

[Stats] OLS/Gauss-Markov assumptions

Answer 35

Basic: linear relationship, X deterministic 1. (y, X) iid 2. E[Y⁴] \< inf, E[X⁴] \< inf (if violate, use L1 cost) 3. X not perfect multicollinear 4. E[e|X] = 0 (if violate, use IV) 5. Var(e) = s²I 6. e ~ Normal 1-4: OLS assumptions 1-5: Gauss-Markov assumptions 1-6: b_est ~ Normal and Y|X ~ Normal

Question 36

[4\* Stats] OLS beta b_est ~ N(?, ?) X_ib_est ~ N(?, ?)

Answer 36

b_est ~ N(b, s²(X'X)^-1), s² = 1/(N-k) Σ e_i² X_ib_est ~ N(X_ib, X_i'Var(b_est )X_i)

Question 37

[Stats] OLS beta group test H₀: b₁ = .. = bK = 0, F = ? H₀: b_K-J+1 = .. = bK = 0, F = ?

Answer 37

H₀: b₁ = .. = bK = 0, F = ((0-SSE)/(K-1)) / (SSE/(N-K)) = (R²/(K-1)) / ((1-R²)/(N-K)) H₀: b_K-J+1 = .. = b_K = 0, F = ((SSE_1..K-J-SSE)/J) / (SSE/(N-K)) = ((R² - R_1..K-J²)/J) / ((1-R²)/(N-K)), SSE_1..K-J and R_1..K-J² from restricted model

Question 38

[Stats] OLS beta OVB 1) y = bX + cZ + e; 2) y = b^OVBX + e b^OVB = b + ?

Answer 38

b^OVB​ = b + c cov(X,Z)/var(X) = b + cd, where Z = dX + e Note: fixed effects in panel regressions partially resolve this, since it removes any omitted variables solely associated with entities or times

Question 39

[Stats] OLS log transform What's dy/dx and what does b represent? y = bx + e y = b ln(x) + e ln(y) = bx + e ln(y) = b ln(x) + e

Answer 39

(y, x): dy/dx = b (y, ln(x)): dy/dx = b 1/x --\> b = dy/(dx/x) (ln(y), x): dy/dx = b e^bx+e = by --\> (dy/y)/dx (ln(y), ln(x)): dy/dx = b 1/x e^bx+e = b y/x --\> (dy/y)/(dx/x)

Question 40

[Stats] WLS and GLS What are solutions when Var(e) = Σ != s²I? If accept inefficient solution, adjust SE.. If not, use alternative estimator..

Answer 40

If accept inefficient solution: NW or HAC (diag only) SE adj Var(b_est) = (X'X)^-1X'ΣX(X'X)^-1 If not, use alternative estimator: GLS / WLS (diag only) y = Xb + e --\> Py = PXb + Pe, where P'P = Σ^-1 --\> y~ = Py, X~ = Px b_est = (X~'X~)^-1X~'y~ = (X'Σ^-1X)^-1X'Σ^-1y Var(b_est) = s²(X~'X~)^-1 = s²(X'Σ^-1X)^-1, where s² = 1/(N-k) (y~-X~b_est)'(y~-X~b_est) = 1/(N-k) (y-Xb)'Σ^-1(y-Xb)

Question 41

[Stats] BLUE What's B, L, U, Estimator?

Answer 41

Best: efficient, Var(statistic_est) = min Var(statistic_i) for all i Linear: statistic_est = Ay Unbiased: E[statistic_est] = statistic