题目410 · 概率
Joint Density and Covariance of Two Uniform Order Statistics
Let $X_1, \ldots, X_n$ be iid $\operatorname{Uniform}(0,1)$. Consider the order statistics $X_{(i)}$ and $X_{(j)}$ with $1 \le i < j \le n$.
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中文题目题目405 · 概率
Joint Distribution of Extremes and the Range
Let $X_1, \ldots, X_n$ be iid $\operatorname{Uniform}(0,1)$. Let $X_{(1)} = \min_i X_i$ and $X_{(n)} = \max_i X_i$.
打开 →题目5481 · 金融与交易
Join Or Improve 1
If you join the current best quote, fill probability is 0.26, size is 120, raw edge per share is 0.011, and rebate is 0.0005. If you improve the quote, fill probability becomes 0.18, size is 120, raw edge per share is 0.017, and rebate is 0.0005. Which action has higher expected
打开 →题目2011 · 数学
Joint Convexity of Size-Time Execution Cost 16
The cost couples size and trading time through a perspective form. Show that P(x,t)=x^2/t + 3 t is convex on the domain t>0.
打开 →题目6011 · 概率
Joint Count of Two Split Streams
A Poisson process at rate $\lambda=30$ per hour is split by independent fair-coin-style labeling into a 'lit' stream (prob $0.2$) and a 'dark' stream (prob $0.8$). Over the next 30 minutes, what is the probability of observing exactly $4$ lit prints and exactly $9$ dark prints?
打开 →题目3525 · 随机过程
Joint Max-End Probability With t = 1, a = 1, b = 0
For standard Brownian motion, what is P(max_{0<=s<=1} W_s >= 1, W_1 <= 0)?
打开 →题目1639 · 统计
Joint MLEs in a Normal Model
Suppose $X_1,\dots,X_9$ are modeled as i.i.d. $N(\mu,\sigma^2)$. From the sample you know that $$\bar X = 5, \qquad \sum_{i=1}^9 (X_i-\bar X)^2 = 18.$$ Find the MLEs of $\mu$ and $\sigma^2$.
打开 →题目3528 · 随机过程
Barrier Needed for a 5% Joint Tail With Endpoint Cap 0
For standard Brownian motion over t = 1, what barrier a gives P(max_{0<=s<=1} W_s >= a, W_1 <= 0) = 0.05?
打开 →题目3529 · 随机过程
Endpoint Cap Needed for a 10% Joint Tail
For standard Brownian motion over t = 1 with barrier a = 1, what endpoint cap b gives P(max_{0<=s<=1} W_s >= 1, W_1 <= b) = 0.10?
打开 →题目2862 · 概率
Factorized Joint MGF Means Independence
Suppose \[ M_{X,Y}(s,t)=\exp\!\left(s+2t+\frac{s^2}{2}+2t^2\right). \] Identify the marginal laws of $X$ and $Y$, and determine whether they are independent.
打开 →题目2848 · 概率
Reading Covariance from a Joint MGF
Suppose \[ M_{X,Y}(s,t)=\exp\!\bigl(2s-t+2s^2+3st+\tfrac52 t^2\bigr). \] Compute $E[X]$, $E[Y]$, $\mathrm{Var}(X)$, $\mathrm{Var}(Y)$, and $\mathrm{Cov}(X,Y)$.
打开 →题目3429 · 数学
Entropy of a Three-Regime Disjoint Alphabet Source
A regime label takes values A, B, C with probabilities 0.2, 0.5, 0.3. Conditional on A the source is deterministic, conditional on B it is uniform over 4 symbols, and conditional on C it is uniform over 2 symbols. Assuming the symbol sets are disjoint across regimes, what is the
打开 →题目056 · 概率
Independent Events Cannot Be Disjoint
Suppose $P(A) = 1/3$ and $P(B) = 1/2$. Can $A$ and $B$ be simultaneously independent and mutually exclusive? Prove your answer.
打开 →题目2536 · 机器学习
Why Softmax Fixes Joint Normalization 16
What does the softmax construction add that one-vs-rest logistic models do not provide automatically?
打开 →题目063 · 概率
A Function of Independent Variables Need Not Be Independent of Its Inputs
Let $X$ and $Y$ be independent $\text{Bernoulli}(1/2)$ random variables. Define $W = \max(X, Y)$. (a) Compute the distribution of $W$. (b) Determine whether $X$ and $W$ are independent by checking all four joint probabilities $P(X = x, W = w)$ for $x, w \in \{0, 1\}$.
打开 →题目240 · 概率
Beta Distribution and the Beta Function from Independent Gammas
Let $X \sim \text{Gamma}(\alpha, 1)$ and $Y \sim \text{Gamma}(\beta, 1)$ be independent. (a) Define $U = \frac{X}{X+Y}$ and $V = X + Y$. Compute the Jacobian of the transformation $(X, Y) \mapsto (U, V)$. (b) Derive the joint PDF of $(U, V)$ and show that $U$ and $V$ are indepe
打开 →题目400 · 概率
Deriving the Fisher F-Distribution from Chi-Squared Variables
Let $X \sim \chi^2(m)$ and $Y \sim \chi^2(n)$ be independent. Define $$F = \frac{X/m}{Y/n}.$$ (a) Using the transformation $(F, W) = \bigl(\frac{nX}{mY},\; Y\bigr)$, compute the Jacobian and derive the joint density $f_{F,W}$. (b) Integrate out $W$ to obtain the marginal PDF of
打开 →题目415 · 概率
Distribution of the Mid-Range for Uniform Samples
Let $X_1, \ldots, X_n$ be iid $\operatorname{Uniform}(0,1)$ with $n \ge 2$. The mid-range is defined as $M = \frac{X_{(1)} + X_{(n)}}{2}$. Using the joint density of $(X_{(1)}, X_{(n)})$, derive the PDF of $M$.
打开 →题目380 · 概率
Distribution of the Ratio of Two Independent Exponentials
Let $X$ and $Y$ be independent $\operatorname{Exp}(1)$ random variables. Define $R = X/Y$. (a) Using the transformation $(R, S) = (X/Y,\, Y)$, compute the joint density $f_{R,S}$ via the Jacobian and then marginalize over $S$ to find the PDF of $R$. (b) Identify $f_R$ as a name
打开 →题目065 · 概率
Mixtures of Independent Distributions Destroy Independence
A biased coin lands heads with probability $1/2$. If heads, set $(X, Y) = (1, 1)$. If tails, draw $X$ and $Y$ independently, each $\text{Bernoulli}(1/2)$. (a) Compute the full joint distribution of $(X, Y)$. (b) Show that $X$ and $Y$ have the same marginal distribution. (c) Are $
打开 →题目389 · 概率
Ratio of Independent Gammas Yields a Beta Distribution
Let $X \sim \operatorname{Gamma}(\alpha, 1)$ and $Y \sim \operatorname{Gamma}(\beta, 1)$ be independent. Using the transformation $(W, S) = \bigl(X/(X+Y),\; X+Y\bigr)$: (a) Compute the Jacobian of the inverse map. (b) Derive the joint density $f_{W,S}$ and marginalize to show $
打开 →题目239 · 概率
Ratio of Independent Standard Normals is Cauchy
Let $Z_1, Z_2$ be independent $N(0,1)$ random variables. Define $R = Z_1 / Z_2$. (a) Write the joint PDF of $(Z_1, Z_2)$ and use the transformation $(Z_1, Z_2) \mapsto (R, Z_2) = (Z_1/Z_2,\, Z_2)$ to derive the joint PDF of $(R, Z_2)$. (b) Integrate out $Z_2$ to obtain the marg
打开 →题目394 · 概率
Ratio of Independent Standard Normals Is Cauchy
Let $X_1, X_2 \sim \text{iid } N(0,1)$. Using the transformation $(Y, V) = (X_1/X_2,\, X_2)$: (a) Derive the joint density $f_{Y,V}(y,v)$. (b) Integrate out $V$ to obtain the marginal PDF of $Y = X_1/X_2$ and identify the distribution.
打开 →题目393 · 概率
Sum of Squares of Two Standard Normals
Let $X_1, X_2 \sim \text{iid } N(0,1)$. Define $R = X_1^2 + X_2^2$. (a) Switching to polar coordinates $(X_1, X_2) = (r\cos\theta, r\sin\theta)$, derive the joint density of $(R, \Theta)$ where $R = X_1^2 + X_2^2$ and $\Theta = \arctan(X_2/X_1)$. (b) Marginalize over $\Theta$ t
打开 →题目058 · 概率
XOR Pairwise Independence Without Mutual Independence
Let $X$ and $Y$ be independent $\text{Bernoulli}(1/2)$ random variables. Define $Z = X \oplus Y$ (where $\oplus$ denotes addition modulo 2). (a) Show that $Z \sim \text{Bernoulli}(1/2)$. (b) Show that any two of $\{X, Y, Z\}$ are independent. (c) Are $X$, $Y$, $Z$ mutually indepe
打开 →题目059 · 概率
All Triples Independent but Quadruple Not
Let $\Omega = \{0, 1, 2, \ldots, 7\}$ with uniform probability $P(\{\omega\}) = 1/8$. Write each $\omega$ in binary as $(b_2, b_1, b_0)$. Define events: $$A = \{\omega : b_0 = 1\}, \quad B = \{\omega : b_1 = 1\}, \quad C = \{\omega : b_2 = 1\}, \quad D = \{\omega : b_0 \oplus b_1
打开 →题目3530 · 随机过程
Barrier Survival Probability for a = 1 Over t = 1
For standard Brownian motion, what is P(max_{0<=s<=1} W_s < 1)?
打开 →题目385 · 概率
Box-Muller Transform: From Uniforms to Independent Normals
Let $U_1, U_2$ be independent $\operatorname{Uniform}(0,1)$ random variables. Define $$Z_1 = \sqrt{-2\ln U_1}\,\cos(2\pi U_2), \qquad Z_2 = \sqrt{-2\ln U_1}\,\sin(2\pi U_2).$$ (a) Compute the Jacobian of the inverse transformation from $(Z_1, Z_2)$ back to $(U_1, U_2)$. (b) Sho
打开 →题目419 · 概率
Conditional Distribution of the Minimum Given the Maximum
Let $X_1, \ldots, X_n$ be iid $\operatorname{Uniform}(0,1)$ with $n \ge 3$. Let $X_{(1)}$ and $X_{(n)}$ denote the minimum and maximum.
打开 →题目376 · 概率
Distribution of the Cube of a Uniform Random Variable
Let $X \sim \operatorname{Uniform}(0,1)$. Using the CDF method, derive the PDF of $Y = X^3$.
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