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中文题目A price follows a driftless random walk whose daily increments are iid with standard deviation 2 bp. By what multiple does the standard deviation of the cumulative move grow when the horizon increases from 1 day to 9 days?
打开 →Suppose $x_t=x_{t-1}+w_t$ with $w_t\sim N(0,1)$, and $y_t=x_t+v_t$ with $v_t\sim N(0,4)$. At time $t-1$ the filtered state is $N(-2,5)$. You observe $y_t=0$. Compute the predicted mean/variance and the updated mean/variance at time $t$.
打开 →Consider the path graph $P_n$ with vertices $\{0, 1, \ldots, n-1\}$ and $n-1$ edges (each of unit resistance), where the walk at interior vertices moves left or right with equal probability, and at the endpoints moves to the unique neighbor. (a) Compute the effective resistance
打开 →A simple random walk moves on the complete graph $K_4$. At each step, the walker moves to one of the $3$ neighbors uniformly at random. (a) Compute the maximum hitting time $t_{\mathrm{hit}} = \max_{u,v} h(u \to v)$. (b) Using Matthews' theorem, which gives $$t_{\mathrm{hit}} \
打开 →A random walk moves on the path graph $P_3$ with vertices $\{1, 2, 3\}$ and edges $\{1{-}2, 2{-}3\}$. At each step, the walker moves to a uniformly random neighbor (so from vertex $2$ it goes to $1$ or $3$ each with probability $\tfrac{1}{2}$, and from vertex $1$ or $3$ it moves
打开 →Consider the complete bipartite graph $K_{2,3}$ with parts $A = \{a_1, a_2\}$ (each of degree $3$) and $B = \{b_1, b_2, b_3\}$ (each of degree $2$). Each edge has unit resistance. (a) Compute the effective resistance $R_{\mathrm{eff}}(a_1, a_2)$ between the two vertices of part
打开 →Consider the cycle graph $C_4$ with vertices $\{0,1,2,3\}$ arranged in a square, and each edge having unit resistance. (a) Compute the effective resistance $R_{\mathrm{eff}}(0, 2)$ between the two diagonally opposite vertices. (b) Using the commute-time identity $C(u,v) = 2m \c
打开 →Consider the $3$-dimensional hypercube graph $Q_3$ (vertices are binary strings of length $3$; edges connect strings differing in exactly one bit). Each edge has unit resistance. (a) Using the symmetry of $Q_3$, compute the effective resistance $R_{\mathrm{eff}}(000, 111)$ betwe
打开 →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 →A simple random walk moves on the complete graph $K_4$ (four vertices, every pair connected). At each step, the walker moves to one of the $3$ neighbors chosen uniformly at random. Starting at a vertex $v$, what is the expected number of steps to return to $v$ for the first time?
打开 →The star graph $S_5$ has a central hub vertex $c$ connected to $4$ leaf vertices $\{\ell_1, \ell_2, \ell_3, \ell_4\}$. A simple random walk at each step moves to a uniformly random neighbor: from the hub, it goes to each leaf with probability $\tfrac{1}{4}$; from any leaf, it goe
打开 →Consider the complete ternary tree of depth $2$: a root vertex $r$ with $3$ children, each of which has $3$ children (leaves), giving $13$ vertices total. A simple random walk moves at each step to a uniformly random neighbor. Starting from a leaf vertex, what is the expected num
打开 →Take the complete graph $K_4$ on vertices $\{1,2,3,4\}$ and remove edge $\{1,4\}$. The resulting graph has $5$ edges, with $d(1)=d(4)=2$ and $d(2)=d(3)=3$. A simple random walk moves at each step to a uniformly random neighbor. Starting from vertex $2$, what is the expected numbe
打开 →A random walk moves on the $3$-dimensional hypercube graph $Q_3$: the $8$ vertices are binary strings of length $3$, and two vertices are adjacent if they differ in exactly one coordinate. At each step, the walker picks one of the $3$ coordinates uniformly at random and flips it.
打开 →Consider the complete bipartite graph $K_{3,3}$ with parts $A = \{a_1, a_2, a_3\}$ and $B = \{b_1, b_2, b_3\}$, where every vertex in $A$ is connected to every vertex in $B$ and vice versa (no edges within a part). A random walk at any vertex moves to each of its $3$ neighbors wi
打开 →A random walk moves on the complete graph $K_5$ (five vertices, every pair connected). At each step, the walker moves to one of the $4$ neighbors chosen uniformly at random. Starting from vertex $u$, what is the expected number of steps to reach a specified vertex $v \neq u$ for
打开 →Take the complete graph $K_4$ on vertices $\{A, B, C, D\}$ and remove edge $A{-}D$, leaving $5$ edges (the "diamond" or "kite" graph). The resulting degrees are $d(A) = d(D) = 2$ and $d(B) = d(C) = 3$. A simple random walk moves at each step to a uniformly random neighbor. (a) S
打开 →Consider the $2 \times 3$ grid graph (ladder graph) with vertices arranged as: $$\begin{matrix} 1 & - & 2 & - & 3 \\ | & & | & & | \\ 4 & - & 5 & - & 6 \end{matrix}$$ Edges connect horizontal and vertical neighbors. A simple random walk moves at each step to a uniformly random ne
打开 →A simple random walk moves on the path graph $P_5$ with vertices $\{0,1,2,3,4\}$ and edges connecting consecutive vertices. At the interior vertices ($1, 2, 3$), the walker moves left or right with equal probability $\tfrac{1}{2}$. At the endpoints ($0$ and $4$), the walker moves
打开 →The Petersen graph has $10$ vertices and $15$ edges; it is $3$-regular, vertex-transitive, and has diameter $2$ (every pair of non-adjacent vertices has exactly one common neighbor, and the graph has girth $5$). A random walk at each step moves to one of the $3$ neighbors uniform
打开 →The wheel graph $W_6$ consists of a central hub $h$ connected to all $5$ vertices of a cycle $C_5$ (so $h$ has degree $5$ and each rim vertex has degree $3$: two cycle neighbors and the hub). A simple random walk moves at each step to a uniformly random neighbor. Starting from a
打开 →A simple random walk moves on the cycle graph $C_8$ (vertices $0, 1, \ldots, 7$ arranged in a circle). At each step, the walker moves clockwise or counterclockwise with equal probability $\tfrac{1}{2}$. Starting at vertex $0$, what is the expected number of steps to reach the ant
打开 →Consider the lazy random walk on the complete graph $K_n$: at each step, the walker stays put with probability $\tfrac{1}{2}$ and moves to a uniformly random neighbor with probability $\tfrac{1}{2}$. (a) Show that the transition matrix has two distinct eigenvalues: $\lambda_0 =
打开 →Consider the lazy random walk on the cycle graph $C_n$: at each step, the walker stays put with probability $\tfrac{1}{2}$, and moves to each of the two neighbors with probability $\tfrac{1}{4}$. The transition matrix has eigenvalues $\lambda_k = \tfrac{1}{2}(1 + \cos(2\pi k / n)
打开 →Consider the graph $G$ on four vertices $\{A, B, C, D\}$ with edges $\{A{-}B,\, A{-}C,\, A{-}D,\, B{-}C\}$, so the degree sequence is $d(A)=3$, $d(B)=2$, $d(C)=2$, $d(D)=1$. A simple random walk moves at each step to a uniformly random neighbor. (a) Find the stationary distribut
打开 →A stationary mean-reverting spread obeys X_(t+1) = 1/2 X_t + epsilon_(t+1), where Var(epsilon_(t+1)) = 4. Starting from the current level, what fraction of the same-horizon random-walk forecast-error variance does the 4-step mean-reverting forecast-error variance represent?
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