题目6043 · 随机过程
Autocorrelation of a Stationary OU Process at a Lag
A stationary OU process has mean-reversion speed kappa = 0.7. What is the autocorrelation between X_t and X_{t+2}?
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中文题目题目2672 · 机器学习
Autocorrelation-Corrected Sample Size
A monthly feature is observed for 60 months and behaves roughly like an AR(1) series with lag-1 autocorrelation $\rho=0.6$. Using the heuristic $n_\text{eff}\approx n(1-\rho)/(1+\rho)$, what is the effective sample size?
打开 →题目680 · 脑筋急转弯
Automaton Visit Repeat 5
A deterministic signal automaton has 81 possible internal states. Starting from any state, after how many visited states must some state have appeared at least twice?
打开 →题目5837 · 金融与交易
Roll Implied Spread From Autocovariance
Under Roll's model, transaction price changes have a first-order serial autocovariance of -0.0009 (in price-squared units). Estimate the implied effective spread.
打开 →题目6038 · 统计
Sign and Size of Lag-1 Autocorrelation
A stationary spread obeys X_(t+1) = -0.4 X_t + epsilon_(t+1) with iid zero-mean shocks. What is the lag-1 autocorrelation of X_t, and what does its sign say about period-to-period dynamics?
打开 →题目1768 · 统计
A Lagged Variable Is Not Automatically a Valid Instrument
Someone proposes using yesterday's order-flow imbalance as an instrument for today's imbalance in a return-impact regression. Why is this not automatically a valid instrument in financial data?
打开 →题目4502 · 数理金融
How Many Candidate Claims Are Automatically Uniquely Priced? 1
In a 4-state market, the traded span is generated by payoffs B0, B1, and B2. Three candidate claims are A = 2 B0 - B1, C = B1 + B2, and D = [1,0,0,0]. How many of A, C, and D are automatically guaranteed a unique arbitrage-free price from the traded market alone?
打开 →题目4504 · 数理金融
How Many Candidate Claims Are Automatically Uniquely Priced? 3
A market's traded span contains F0 and F1 but not the full state space. Candidate claims are M = 3 F0 - 0.5 F1, N = [1,0,1,0], and P = [0,1,0,1]. How many of M, N, and P are automatically guaranteed unique prices?
打开 →题目2419 · 机器学习
Why Low Bias Is Not Automatically Desirable
Why is 'lower bias' not automatically a sufficient argument for preferring one model over another?
打开 →题目4503 · 数理金融
How Many Candidate Claims Are Automatically Uniquely Priced? 2
In a 5-state market, traded payoffs span H1, H2, and H3. Candidate claims are U = H1 + 2 H3, V = H2 - H1, and W = [0,1,0,0,0]. How many of U, V, and W are guaranteed uniquely priced just from attainability?
打开 →题目4505 · 数理金融
How Many Candidate Claims Are Automatically Uniquely Priced? 4
In a 4-state market, traded payoffs span G0, G1, and G2. Candidate claims are A = G0 + G1, B = 2 G2 - G1, C = G0 - G2, and D = [0,0,0,1]. How many of A, B, C, and D are guaranteed uniquely priced from the given information?
打开 →题目2861 · 概率
Why the Difference of Two Copies Is Automatically Symmetric
Let $X$ and $Y$ be i.i.d. with characteristic function $\phi(u)$. Show that the characteristic function of $D=X-Y$ is $|\phi(u)|^2$, and conclude that $D$ is symmetric about $0$.
打开 →题目6040 · 统计
Two-Period Variance Ratio of an AR(1)
Returns are generated by a stationary AR(1) with autoregressive coefficient 0.5. The Lo-MacKinlay variance ratio at lag 2 is VR(2) = Var(r_t + r_(t+1)) / (2 Var(r_t)). Compute VR(2) and state whether it signals momentum or mean reversion.
打开 →题目268 · 概率
ABCA in a Four-Symbol Stream
Each symbol of an iid stream is chosen uniformly from $\{A,B,C,D\}$. Compute the expected waiting time until $ABCA$ first appears.
打开 →题目274 · 概率
Either 123 or 321 on a Fair Die
A fair six-sided die is rolled repeatedly. Let $T$ be the first time that either $1,2,3$ or $3,2,1$ appears as a consecutive length-3 block. Compute $E[T]$.
打开 →题目264 · 概率
Expected Time Until ABC or CBA
A symbol stream is iid and uniform on $\{A,B,C\}$. What is the expected number of symbols until either $ABC$ or $CBA$ first appears?
打开 →题目271 · 概率
Finish 1-2-3 After Already Rolling 1,2
A fair six-sided die is rolled repeatedly. You already know that the current observed suffix is exactly $1,2$. Starting from here, what is the expected additional number of rolls until $1,2,3$ first appears?
打开 →题目269 · 概率
Finish ABCA After Current Suffix ABC
An iid stream over $\{A,B,C,D\}$ is uniform. Suppose the current observed suffix is exactly $ABC$. From this point onward, what is the expected additional number of symbols until $ABCA$ first appears?
打开 →题目263 · 概率
First Hit of ABA or BAA
An iid stream over $\{A,B,C\}$ is uniform. Let $T$ be the first time that either $ABA$ or $BAA$ appears. Compute $E[T]$.
打开 →题目265 · 概率
Nonuniform Stream Waiting for AABA
An iid source emits $A,B,C$ with probabilities $ rac{1}{2}, rac{1}{3}, rac{1}{6}$ respectively. Find the expected waiting time until $AABA$ first appears.
打开 →题目267 · 概率
Nonuniform Stream Waiting for ABAC
An iid source emits $A,B,C$ with probabilities $ rac{1}{2}, rac{1}{3}, rac{1}{6}$ respectively. Find the expected waiting time until $ABAC$ first appears.
打开 →题目266 · 概率
Nonuniform Stream Waiting for ABBA
An iid source emits $A,B,C$ with probabilities $ rac{1}{2}, rac{1}{3}, rac{1}{6}$ respectively. What is the expected number of emitted symbols until $ABBA$ first appears?
打开 →题目257 · 概率
Overlap Penalty for HTHT
A fair coin is flipped repeatedly. Let $T$ be the waiting time until $HTHT$ first appears. Compute $E[T]$.
打开 →题目273 · 概率
Return-and-Overlap Pattern 1-2-1 on a Die
A fair six-sided die is rolled repeatedly. Find the expected waiting time until the consecutive pattern $1,2,1$ first appears.
打开 →题目270 · 概率
Rolls Until 1-2-3 Appears
A fair six-sided die is rolled repeatedly. What is the expected number of rolls until the consecutive pattern $1,2,3$ first appears?
打开 →题目272 · 概率
Three Consecutive Ones on a Die
A fair six-sided die is rolled repeatedly. Let $T$ be the waiting time until three consecutive 1s first appear. Compute $E[T]$.
打开 →题目262 · 概率
Waiting for ABAB in a Uniform ABC Stream
Each symbol in an iid stream is chosen uniformly from $\{A,B,C\}$. Find the expected waiting time until $ABAB$ first appears.
打开 →题目275 · 概率
Waiting for HHTH
A fair coin is flipped repeatedly. What is the expected number of flips until $HHTH$ first appears?
打开 →题目1931 · 统计
Effective Sample Under Serial Dependence 1
A return series has 240 observations and lag-1 autocorrelation 1/5. Using the heuristic n_eff = n * (1-rho)/(1+rho), what is the effective sample size?
打开 →题目4813 · 数理金融
HJM First Diagnostic 23
Before treating HJM flexibility as an automatic advantage, what operational cost should you state first?
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