题目398 · 概率
Additivity of Chi-Squared Distributions via MGF
Let $X \sim \chi^2(m)$ and $Y \sim \chi^2(n)$ be independent. Using moment-generating functions, prove that $X + Y \sim \chi^2(m + n)$.
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中文题目题目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3454 · 数学
KL Between Signal Distributions Under the Two Regimes
A binary signal Y has P(Y=1|R=1) = 0.8 and P(Y=1|R=0) = 0.3. What is KL(P(Y|R=1) || P(Y|R=0)) in bits?
打开 →题目3436 · 数学
Why Uniform Distributions Maximize Entropy Under a Support Constraint
Why does spreading probability mass more evenly over a fixed finite support raise entropy?
打开 →题目067 · 概率
Identical Marginals Do Not Imply Independence
Three cards are labeled $(0,0)$, $(0,1)$, and $(1,0)$. One card is drawn uniformly at random. Let $X$ be the first number and $Y$ the second number on the drawn card. (a) Compute $P(X = 0)$, $P(X = 1)$, $P(Y = 0)$, and $P(Y = 1)$. (b) Are the marginal distributions of $X$ and $Y$
打开 →题目245 · 概率
Maximum Entropy Property of the Normal Distribution
The differential entropy of a continuous random variable $X$ with PDF $f$ is $h(X) = -\int_{-\infty}^{\infty} f(x) \ln f(x)\, dx$. (a) Among all continuous distributions on $\mathbb{R}$ with mean $\mu$ and variance $\sigma^2$, use the method of Lagrange multipliers to show that
打开 →题目2391 · 数学
95% Half-Width From Sample Volatility and Path Count 21
A Monte Carlo estimator has sample standard deviation 5 across n=400 paths. Using the normal approximation, what is the 95% half-width?
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399 · 概率
Absolute Value of a Standard Normal: The Half-Normal Distribution
Let $X \sim N(0,1)$ and define $Y = |X|$. (a) Derive the PDF of $Y$ using the CDF method (note that $Y = |X|$ is not monotone). (b) Compute $E[Y]$ and $\operatorname{Var}(Y)$.
打开 →题目386 · 概率
Affine Transformation of a Normal Random Variable
Let $X \sim N(\mu, \sigma^2)$ with $a \neq 0$ and $b \in \mathbb{R}$. Using the Jacobian formula, show that $Y = aX + b$ is normally distributed and state its parameters.
打开 →题目468 · 概率
Aggregate Insurance Claims via the CLT
An insurer has $300$ independent policyholders. Each policyholder files a $\mathrm{Poisson}(3)$ number of claims per year. Let $T = \sum_{i=1}^{300} N_i$ be the total number of claims. Using the CLT, approximate $P(T > 960)$. You may use $\Phi(2.00) \approx 0.9772$.
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: $$A = \{\omega : b_0 = 1\}, \quad B = \{\omega : b_1 = 1\}, \quad C = \{\omega : b_2 = 1\}, \quad D = \{\omega : b_0 \oplus b_1
打开 →题目432 · 概率
Asymmetric Penalties in an Exponential Race
Two independent alarms go off at $\operatorname{Exp}(4)$ and $\operatorname{Exp}(6)$ times respectively. If alarm 1 fires first you pay $\$3$; if alarm 2 fires first you pay $\$5$. After the first alarm fires, the remaining alarm is reset (memoryless restart) and you pay an addit
打开 →题目470 · 概率
Asymptotic Distribution of the Sample Median
Let $X_1, \ldots, X_n$ be i.i.d.\ $\mathrm{Uniform}(0,1)$ and let $M_n$ denote the sample median (the middle order statistic for odd $n$, or the average of the two middle values for even $n$). The asymptotic theory of order statistics gives $$\sqrt{n}\,(M_n - m) \xrightarrow{d}
打开 →题目5994 · 概率
At Least Two Arrivals
Defaults in a small credit book occur as a Poisson process with rate 8 per year. What is the probability that at least 2 defaults occur in the next 3 months? Give a decimal to three places.
打开 →题目221 · 概率
Bernoulli Distribution: Moments and MGF
Let $X \sim \text{Bernoulli}(p)$, so $P(X=1)=p$ and $P(X=0)=1-p$ where $0 < p < 1$. (a) Compute $E[X]$ and $E[X^2]$ directly from the PMF. Hence find $\text{Var}(X)$. (b) Derive the moment-generating function $M_X(t) = E[e^{tX}]$ and verify that $M_X'(0) = E[X]$ and $M_X''(0) =
打开 →题目454 · 概率
Berry-Esseen Bound for a Skewed Bernoulli Sum
Let $X_1, X_2, \ldots, X_n$ be i.i.d.\ $\operatorname{Bernoulli}(p)$ with $p = 0.01$ and $n = 10{,}000$. Define $S_n = \sum_{i=1}^{n} X_i$. **(a)** Using the CLT, approximate $P(S_n \le 80)$. **(b)** The Berry-Esseen theorem states that $\sup_x |P(Z_n \le x) - \Phi(x)| \le \fra
打开 →题目465 · 概率
Berry-Esseen Bound for a Sum of Uniform Random Variables
Let $U_1, \ldots, U_n$ be i.i.d.\ $\mathrm{Uniform}(0,1)$ and $S_n = \sum_{i=1}^n U_i$. The Berry-Esseen theorem states $$\sup_x \left|P\!\left(\frac{S_n - n/2}{\sigma\sqrt{n}} \le x\right) - \Phi(x)\right| \le \frac{C\,\rho}{\sigma^3 \sqrt{n}},$$ where $\sigma^2 = \mathrm{Var}(U
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413 · 概率
Beta Distribution of the k-th Uniform Order Statistic
Let $X_1, \ldots, X_n$ be iid $\operatorname{Uniform}(0,1)$. Derive that the $k$-th order statistic $X_{(k)}$ has the $\operatorname{Beta}(k, n-k+1)$ distribution.
打开 →题目230 · 概率
Beta Distribution: Derivation, Moments, and Connections
Consider the Beta distribution with PDF $f(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1}$ for $x \in (0,1)$. (a) Verify that $f$ integrates to 1 by showing $\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\,dx = \frac{\Gamma(\alpha)\Gamma(\beta)}{
打开 →题目207 · 概率
Binomial Moments via the Moment Generating Function
Let $X \sim \text{Binomial}(n, p)$. (a) Derive the moment generating function $M_X(t) = E[e^{tX}]$ in closed form. (b) By differentiating $M_X(t)$ and evaluating at $t = 0$, find $E[X]$ and $E[X^2]$, and hence $\text{Var}(X)$.
打开 →题目471 · 概率
Binomial Tail with Continuity Correction
A fair coin is flipped $n = 144$ times. Let $S$ be the number of heads. **(a)** Using the CLT (without continuity correction), approximate $P(S \ge 80)$. **(b)** Repeat with the continuity correction. You may use $\Phi(1.33) \approx 0.9082$ and $\Phi(1.25) \approx 0.8944$.
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453 · 概率
Call Center Overflow via Poisson CLT
A call center receives calls according to a Poisson process with rate $\lambda = 4$ calls per minute. The center operates for an $8$-hour shift ($480$ minutes). The center can handle at most $2000$ calls per shift before service quality degrades. Using a suitable normal approxim
打开 →题目188 · 概率
Capped Weak Compositions
Seven indistinguishable jobs are split across four labeled servers. How many occupancy vectors are possible if no server is allowed to receive more than three jobs?
打开 →题目231 · 概率
CDF and Tail Probability of the Exponential Distribution
Let $X \sim \text{Exponential}(\lambda)$ with PDF $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$. (a) Derive the CDF $F(x) = P(X \leq x)$. (b) If $\lambda = \frac{1}{2}$, compute $P(X > 3)$.
打开 →题目421 · 概率
CDF of the Minimum of Four Uniforms
Let $X_1, X_2, X_3, X_4$ be independent $\operatorname{Uniform}(0,1)$ random variables. Derive the CDF and PDF of $X_{(1)} = \min(X_1, X_2, X_3, X_4)$.
打开 →题目430 · 概率
Characterization of Memorylessness and the Residual Life Paradox
Part (a): Let $X$ be a continuous, positive random variable satisfying $P(X > s + t \mid X > s) = P(X > t)$ for all $s, t \geq 0$. Prove that $X$ must be exponentially distributed. Part (b): A lightbulb's lifetime $L$ has CDF $F(t) = 1 - \frac{1}{2}e^{-t} - \frac{1}{2}e^{-3t}$ f
打开 →题目235 · 概率
Chi-Squared Distribution from First Principles
Let $Z_1, \ldots, Z_n$ be i.i.d. $N(0,1)$ random variables and define $Q = \sum_{i=1}^n Z_i^2$. (a) Derive the PDF of $Z_1^2$ using the change-of-variables technique for transformations of continuous random variables. (b) Show that $Z_1^2 \sim \text{Gamma}\left(\frac{1}{2}, \fr
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