题目2860 · 概率
Characteristic Function of a Uniform Return Shock
Let $X\sim\mathrm{Uniform}[-a,a]$. Compute its characteristic function and recover $\mathrm{Var}(X)$ from the transform.
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中文题目题目193 · 概率
Conditional Expected Occupancy of a Nonempty Urn
Four distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Given that urn 1 is nonempty, what is the expected number of balls in urn 1? Give an exact fraction.
打开 →题目2898 · 概率
Directed Cycle with Clockwise Drift
On the directed cycle with states $\{0,1,\dots,n-1\}$, from each state the chain moves clockwise with probability $p$ and counterclockwise with probability $1-p$. Show that the stationary distribution is uniform.
打开 →题目2964 · 概率
Expected Area of the Rectangle to the Origin
A random point $(X,Y)$ is uniform in the unit square. Consider the axis-aligned rectangle with opposite corners at $(0,0)$ and $(X,Y)$. What is the expected area of that rectangle?
打开 →题目5323 · 金融与交易
Expected Carry Needed To Hit VaR Target
A trading book's 1-day PnL is modeled as normal with mean mu and standard deviation 9 million. Risk reports define delta-normal VaR by VaR = z_alpha*sigma - mu. If z_0.95=1.645 and the desk wants reported 95% VaR to be exactly 12 million, what daily expected PnL mu is required?
打开 →题目2793 · 脑筋急转弯
Expected Chord Length on the Unit Circle
Two points are chosen independently and uniformly on the unit circle. What is the expected length of the chord connecting them?
打开 →题目550 · 概率
Expected Cover Time of the Cycle C₆
A simple random walk moves on the cycle graph $C_6$ (vertices $0, 1, \ldots, 5$). At each step, the walker moves clockwise or counterclockwise with equal probability. Starting at vertex $0$, what is the expected number of steps to visit all $6$ vertices (the expected cover time)?
打开 →题目5820 · 金融与交易
Expected Daily PnL From Volume And Fill Rate
The product trades 2,000,000 shares per day. You are at the touch on enough of that flow that you fill 4 percent of daily volume passively, capturing a net edge of 0.0015 per share after all costs. What is your expected daily PnL?
打开 →题目2949 · 概率
Expected Distance to the Nearest Vertical Side
A random point $(X,Y)$ is uniform on the unit square. Compute the expected distance from the point to the nearer of the two vertical sides.
打开 →题目5531 · 金融与交易
Expected Ending Inventory 1
You start long 200 shares. If you keep both sides on, expected bid-fill probability is 0.24, ask-fill probability is 0.16, and each fill size is 50. If you instead turn the bid off and only leave the ask quote live, what expected ending inventory do you get under each policy, and
打开 →题目3547 · 随机过程
Expected Exit Time From [-2, 4] Started at 1
For Brownian motion started at x = 1 inside [-2, 4], what is the expected exit time from the interval?
打开 →题目3546 · 随机过程
Expected Exit Time From [0, 5] Started at 2
For Brownian motion started at x = 2 inside [0, 5], what is the expected exit time from the interval?
打开 →题目5426 · 金融与交易
Expected Fair Value Across Scenarios 1
Your fast book model gives three possible fair values: 99.6 with p=0.2, 100 with p=0.5, 100.9 with p=0.3. If you quote around the expected fair value, what number should you use?
打开 →题目5427 · 金融与交易
Expected Fair Value Across Scenarios 2
Your fast book model gives three possible fair values: 49.2 with p=0.25, 50.1 with p=0.45, 50.7 with p=0.3. If you quote around the expected fair value, what number should you use?
打开 →题目5821 · 金融与交易
Expected Fills Over Many Rounds
You post the same two-sided quote in each of 8 independent rounds. In every round the quote fills with probability 0.35, and there is no inventory limit so fills do not interact. What is the expected number of fills over the 8 rounds?
打开 →题目424 · 概率
Expected Gap Between the Two Largest Uniforms
Let $X_1, \ldots, X_6$ be iid $\operatorname{Uniform}(0,1)$. Find the expected gap between the largest and second-largest values: $E[X_{(6)} - X_{(5)}]$.
打开 →题目551 · 概率
Expected Occupied Labels 1
10 independent packets are hashed uniformly into 8 labels. What is the expected number of labels hit at least once?
打开 →题目171 · 概率
Expected Pair Count With One Holiday Twice as Likely
On a 366-point calendar, one special holiday has probability 2/366 and each of the other 364 days has probability 1/366. For n=30 independent birthdays, what is the expected number of unordered matching pairs?
打开 →题目5786 · 金融与交易
Expected PnL Of Skewing To Offload
You are long 100 lots. Skewing the ask down attracts an expected sell of 60 lots this period, each lot offloaded at +0.08 of edge versus your reservation price. The 40 lots that remain carry an expected holding cost of 0.15 per lot. What is the expected PnL of the skew policy thi
打开 →题目5810 · 金融与交易
Expected Profit Over N Quotes
Over a session you post 5000 quotes. Each quote has a fill probability of 0.08, and each fill earns a net edge of 0.25. What is your expected total PnL for the session?
打开 →题目5988 · 概率
Expected Sample Size of a Sequential Drift Test
A sequential test accumulates i.i.d. log-likelihood increments $X_1,X_2,\dots$ with $E[X_i]=0.25$. The test stops at $N=\min\{n: |S_n|\ge 3\}$ where $S_n=\sum_{i=1}^n X_i$, and it is given that $E[N]<\infty$ and that the expected stopped statistic is $E[S_N]=2.0$ (reflecting that
打开 →题目556 · 概率
Expected Singleton Labels 1
8 independent arrivals are assigned uniformly to 10 labels. What is the expected number of labels that receive exactly one arrival?
打开 →题目2797 · 脑筋急转弯
Expected Squared Distance in the Unit Disk
Two points are chosen independently and uniformly from the unit disk. What is the expected squared Euclidean distance between them?
打开 →题目5990 · 概率
Expected Time to the Third Arrival
Orders arrive at a matching engine as a Poisson process with rate 4 per minute. What is the expected time, in seconds, until the 3rd order arrives (measured from time 0)?
打开 →题目5826 · 金融与交易
Expected Wait Until First Fill
You repost the same quote every round; each round it fills independently with probability 0.2. Rounds continue indefinitely until the first fill. What is the expected number of rounds until (and including) the first fill?
打开 →题目3549 · 随机过程
Lower Barrier Needed for Expected Exit Time 12
Brownian motion starts at x = 2 with upper barrier 5 and lower barrier a. What a makes the expected exit time equal to 12?
打开 →题目5927 · 概率
Minimize the Expected Rank
Three items arrive in random order; after each you learn its rank relative to those seen so far and must accept or reject irrevocably (the last is forced). Instead of trying to get the single best, you want to minimize the expected absolute rank of the item you finally accept (ra
打开 →题目220 · 概率
Poisson Limit of the Binomial via Characteristic Functions
Let $X_n \sim \text{Binomial}(n, \lambda/n)$ for fixed $\lambda > 0$ and $n > \lambda$. (a) Write down the characteristic function $\varphi_{X_n}(t) = E[e^{itX_n}]$ in closed form. (b) Show that $\lim_{n \to \infty} \varphi_{X_n}(t) = e^{\lambda(e^{it} - 1)}$ for every $t \in \
打开 →题目2853 · 概率
Poisson to Normal via Centered Characteristic Functions
Let $N_\lambda\sim\mathrm{Poisson}(\lambda)$. Show that \[ \frac{N_\lambda-\lambda}{\sqrt{\lambda}}\Rightarrow N(0,1) \quad\text{as }\lambda\to\infty \] by working directly with characteristic functions.
打开 →题目2851 · 概率
Rademacher CLT through Characteristic Functions
Let $X_1,X_2,\dots$ be i.i.d. with $P(X_i=1)=P(X_i=-1)=1/2$. Show, using characteristic functions, that \[ \frac{X_1+\cdots+X_n}{\sqrt n}\Rightarrow N(0,1). \]
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