2.4 For the general two-state chain with transition matrix P: a b (72) P a and initial distribution a = (a₁, a₂), find the following: (a) the two-step transition matrix (b) the distribution of X₁
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2.4
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- For the transition probability matrix P = 1 2 3 4 5 1 0.2 0.1 0.15 0 0.55 2 0 1 0 0 0 3 0.35 0.2 0.2 0.1 0.15 4 0 0 0 1 0 5 0.25 0.2 0.15 0.25 0.15 (a) Rewrite P in the canonical form, clearly identifying R and Q. (b) For each state, i, calculate the mean number of times that the process is in a transient state j, given it started in i. (c) For each state i, find the mean number of transitions before the process hits an absorbing state, given that the process starts in a transient state i. (d) For each state i, find the probability of ending in each of the absorbing states.Simplify the following Boolean functions, using K-maps: F (w, x, y, z)=Σ(11, 12, 13, 14, 15)In Sammon mapping, if the mapping is linear, namely, g(x)W) = Wt x, how can W that minimizes the Sammon stress be calculated?
- Consider a three-qubit quantum system, with each qubit described by the computational basis states |0⟩ and |1⟩. The system is in the following quantum state: (a) Find the reduced density matrix ρA for the first qubit (from the left) by tracing out the other two qubits. (b) Show that the state |Ψ⟩ is entangled by proving that it cannot be written as a product state of the form |ψA⟩ ⊗ |ψBC ⟩.Prove that the following instance of PCP does not have a match [aba, bb], [ab, a], [bab, aa], [ab, ba]Consider the Markov chain with three states,S={1,2,3}, that has the following transition matrix P= Draw the state transition diagram for this chain. If we know P(X1=1) =P(X1=2) =1/4, find P(X1=3, X2=2,X3=1)
- 3.6 a. Use the hybrid (binary/bipolar) form of Hebb rule learning as described in Example 3.6 to find the weight matrix for the associative memory network based on the following binary input-output vector pairs: s=0100 0 (=01 0 =01 001 @@= 0 s3) =0 1.0 0 t3)=0 1 s@=0 110 H4) =@ 1) b. Using the unit step function (with threshold 0) as the output units’ activation function, test the response of your network on each of the input patterns. Describe the results obtained. c. Test the response of your network on various combinations of input patterns with “‘mistakes” or ‘‘missing-data’ (as in Example 3.8). Discuss the results you ob- serve.In which case are Sobol coefficients not a good measure of sensitivity ? a. if the model can be well approximated with a gPC expansion b. If many of the the higher order moments of the output are large c. if the input random variables are dependentHere is the Berkeley Madonna code: {Top model} {Reservoirs} d/dt (Q) = + Stimulus - Imemb INIT Q = -65/cap {Flows} Stimulus = Intensity*SquarePulse(3,.5) {at t=3 of 0.5 duration} Imemb = IL+IK+INa {Functions} Intensity = 100 {microamps} cap = 1 E = Q/cap {Submodel "INa_"} {Functions} ENa = 50 INa = gNa*(E-ENa) GNaMax = 120 gNa = GNaMax*m*m*m*h {Submodel "m_gates"} {Reservoirs} d/dt (m) = + m_prod - m_decay INIT m = am/(am+bm) {Flows} m_prod = am*(1-m) m_decay = bm*m {Functions} am = 0.1*(E+40)/(1-exp(-(E+40)/10)) bm = 4*exp(-(E+65)/18) {Submodel "h_gates"} {Reservoirs} d/dt (h) = + h_prod - h_decay INIT h = ah/(ah+bh) {Flows} h_prod = ah*(1-h) h_decay = bh*h {Functions} ah = 0.07*exp(-(E+65)/20) bh =…
- Consider a system with input x(t) and output y(t) . The relationship between input and output is y(t) = x(t)x(t − 2) a. Is the system causal or non-causal?b. Determine the output of system when input is Aδ(t) , where A is any real orcomplex number?Consider the following time series：Yt=Yt−1−0.25Yt−2+wt−0.5wt−1，wt is a white noise process with variance σw^2. Simulate a sample from the reduced model and calculate the sample ACF and PACF. Plot the sample ACF and theoretical ACF on one graph, and the sample PACF and the theoretical PACF on the other graph. Show the two graphs on one figure. (Please provide R codes).Given the complement of a graph G is a graph G' which contains all the vertices of G, but for each unweighted edge that exists in G, it is not in G', and for each possible edge not in G, it is in G'. What logical operation and operand(s) can be applied to the adjacency matrix of G to produce G'? AND G's adjacency matrix with 0 to produce G' XOR G's adjacency matrix with 0 to produce G' XOR G's adjacency matrix with 1 to produce G' AND G's adjacency matrix with 1 to produce G'