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中文题目Let $A$, $B$, $C$ be events. (a) Prove that if $A$, $B$, $C$ are mutually independent, then $A$ is independent of $B \cap C^c$. (b) Now let $\Omega = \{1,2,3,4\}$ with uniform probability, $A = \{1,2\}$, $B = \{1,3\}$, $C = \{1,4\}$. Verify that $A$, $B$, $C$ are pairwise indepen
打开 →A four-card deck contains $A\spadesuit$, $A\heartsuit$, $K\spadesuit$, $K\heartsuit$. One card is drawn uniformly at random. Let $R$ = 「the card is an Ace」 and $S$ = 「the card is a Spade.」 (a) Compute $P(R)$, $P(S)$, $P(R \cap S)$ and verify that $R$ and $S$ are independent. (b)
打开 →Let $A$, $B$, $C$ be events with $A \perp\!\!\perp B$ and $A \perp\!\!\perp C$. (a) Prove that if $A$, $B$, $C$ are mutually independent, then $A \perp\!\!\perp (B \cup C)$. (b) Now let $\Omega = \{1,2,3,4\}$ with uniform probability, $A = \{1,2\}$, $B = \{1,3\}$, $C = \{1,4\}$.
打开 →Let $\Omega = \{1,2,3,4\}$ with uniform probability. Define $A = \{1,2\}$ and $B = \{1,3\}$. (a) Verify that $A$ and $B$ are independent. (b) Let $A_1 = \{1\} \subseteq A$. Determine whether $A_1$ and $B$ are independent. (c) State a general principle: if $A \perp\!\!\perp B$ and
打开 →Let $\Omega = \{1,2,3,4\}$ with uniform probability. Define $A = \{1,2\}$, $B = \{1,3\}$, $C = \{3,4\}$. (a) Verify that $A$ and $B$ are independent. (b) Verify that $B$ and $C$ are independent. (c) Are $A$ and $C$ independent? What does this say about transitivity of independenc
打开 →A coin is chosen at random: with probability $1/2$ it is fair ($p = 1/2$) and with probability $1/2$ it is biased ($p = 1$, always heads). Let $A$ be the event that the first flip is heads and $B$ the event that the second flip is heads. Show that $A$ and $B$ are conditionally in
打开 →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 →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 →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 →Let $\Omega$ consist of all binary strings of length $4$ with an even number of $1$s, each equally likely: $$\Omega = \{0000,\, 0011,\, 0101,\, 0110,\, 1001,\, 1010,\, 1100,\, 1111\}.$$ Define events $A_i = \{\omega \in \Omega : \omega_i = 1\}$ for $i = 1,2,3,4$. (a) Show that $
打开 →Let the sample space be $\Omega = \{1, 2, 3, 4\}$ with uniform probability $P(\{i\}) = 1/4$. Define $A = \{1, 2\}$, $B = \{1, 3\}$, $C = \{1, 4\}$. Show that $A$, $B$, $C$ are pairwise independent but not mutually independent.
打开 →Roll two fair six-sided dice. Define events $A = \{\text{sum is } 7\}$ and $B = \{\text{first die shows an even number}\}$. Are $A$ and $B$ independent? Prove your answer by computing $P(A)$, $P(B)$, and $P(A \cap B)$.
打开 →Let $\Omega = \{0,1,\ldots,15\}$ with uniform probability, representing all $4$-bit binary strings $b_1 b_2 b_3 b_4$. Define four events: $A_i = \{\omega \in \Omega : b_i(\omega) = 1\}$ for $i = 1,2,3$, and $A_4 = \{\omega \in \Omega : b_1 \oplus b_2 \oplus b_3 = 1\}$ where $\opl
打开 →Let $N \sim \text{Poisson}(\lambda)$. Each of the $N$ events is independently classified as type 1 with probability $p$ and type 2 with probability $1 - p$. Let $N_1$ and $N_2$ denote the counts of type 1 and type 2 events, respectively. (a) Derive the marginal distribution of $
打开 →Let $\Omega = \{1,2,\ldots,8\}$ with uniform probability. Define events: $$A = \{1,2,3,4\}, \quad B = \{1,2,3,5\}, \quad C = \{1,4,6,7\}.$$ (a) Show that $P(A \cap B \cap C) = P(A)P(B)P(C)$. (b) Check whether each pair $(A,B)$, $(A,C)$, $(B,C)$ is independent. (c) What does this
打开 →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 →Let $X$ be uniformly distributed on $\{-1, 0, 1\}$. Define $Y = X^2$. Compute $\text{Cov}(X, Y)$ and determine whether $X$ and $Y$ are independent.
打开 →A 2x2 contingency table of outcomes is $$\begin{pmatrix}30 & 20\\ 10 & 40\end{pmatrix}.$$ Test independence using the classical chi-square test.
打开 →You have only a few hundred labeled observations, but domain knowledge gives a plausible class-conditional structure and you also have many unlabeled feature vectors. Would you start with a generative or a discriminative model first?
打开 →Why does making two Brownian linear combinations uncorrelated automatically make them independent in these questions?
打开 →Let $A$ be an event in a probability space. (a) Write the independence condition $P(A \cap A) = P(A) \cdot P(A)$ and deduce which values of $P(A)$ satisfy it. (b) On $\Omega = \{1,2,3,4\}$ with uniform probability, verify your answer by testing $A = \{1\}$, $A = \{1,2\}$, $A = \e
打开 →Let $A$ and $B$ be independent events with $P(A \cup B) = 1$. (a) Using the independence condition and inclusion-exclusion, prove that $(1 - P(A))(1 - P(B)) = 0$. (b) What does this force about $P(A)$ and $P(B)$? (c) On $\Omega = \{1,2,3,4\}$ with uniform probability, find events
打开 →Let $\Omega = \{0, 1, 2, \ldots, 11\}$ with uniform probability $P(\{k\}) = 1/12$ for each $k$. Define three events based on divisibility: $A = \{k \in \Omega : 2 \mid k\}$ (even numbers), $B = \{k \in \Omega : 3 \mid k\}$ (multiples of $3$), $C = \{k \in \Omega : 4 \mid k\}$ (mu
打开 →A permutation $\sigma$ of $\{1,2,3,4\}$ is chosen uniformly at random (each of the $4! = 24$ permutations equally likely). For $i = 1,2,3,4$, define the event $A_i = \{\sigma(i) = i\}$ (element $i$ is a fixed point). (a) By counting, show that $P(A_i) = 1/4$ for every $i$, and $P
打开 →Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A$ be an event with $P(A) = 0$. (a) Prove that $A$ is independent of every event $B \in \mathcal{F}$. (b) Let $\Omega = \{1,2,3,4,5,6\}$ with uniform probability. Set $A = \emptyset$ and $B = \{1,2,3\}$. Verify the in
打开 →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 →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 →Let $X \sim \text{Poisson}(\lambda)$ and $Y \sim \text{Poisson}(\mu)$ be independent. (a) Derive the MGF of a $\text{Poisson}(\lambda)$ random variable. (b) Using MGFs, prove that $X + Y \sim \text{Poisson}(\lambda + \mu)$. (c) A call centre receives calls from two independent
打开 →Let $A$ and $B$ be independent events. Prove that $A^c$ and $B^c$ are also independent, i.e., $P(A^c \cap B^c) = P(A^c) \cdot P(B^c)$.
打开 →Two independent random points are drawn uniformly from the unit square. What is the probability that both coordinate differences satisfy \[ |X_1-X_2|<\frac12,\qquad |Y_1-Y_2|<\frac12? \]
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