C877 Mathematical Modeling and Applications
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Free C877 Mathematical Modeling and Applications Questions
In difference equations, the solution to y_{n+1} = 1.5 y_n with y_0 = 100 is:
- y_n = 100 × 1.5^n
- y_n = 100 × (1.5^n − 1)
- y_n = 100 / 1.5^n
- y_n = 150 n
Explanation
This is a linear nonhomogeneous recurrence with no constant term, so geometric solution y_n = A r^n. Back-substitute: A r^{n+1} = 1.5 A r^n → r = 1.5. Initial condition y_0 = 100 → A = 100. Thus y_n = 100 × 1.5^n, which is discrete exponential growth, exactly analogous to continuous y(t) = 100 e^{rt} with r = ln(1.5).
Correct Answer Is:
y_n = 100 × 1.5^n
The Gompertz growth model is used for tumor growth. The DE is:
- dN/dt = −c N ln(N/K)
- dN/dt = r N (1 − N/K)
- dN/dt = r N
- dN/dt = r e^{-t}
Explanation
Gompertz model assumes growth rate decreases logarithmically with size: dN/dt = −c N ln(N/K), equivalent to r(t)N where r(t) = c ln(K/N). This produces an asymmetric S-curve common in tumors, actuarial data, and some populations, unlike symmetric logistic. Solution N(t) = K exp(−α e^{-ct}).
Correct Answer Is:
dN/dt = −c N ln(N/K)
The Akaike Information Criterion (AIC) for model selection is: AIC = −2 ln(L) + 2k
- AIC = 2k − 2 ln(L)
- AIC = n ln(RSS/n) + 2k
- AIC = −2 ln(L) + 2k
- AIC = deviance + 2k
Explanation
AIC = −2 ln(maximum likelihood) + 2k penalizes complexity. Lower AIC is better. For Gaussian errors it simplifies to n ln(RSS/n) + 2k. In C877 you compute AIC for logistic vs Gompertz vs Richards growth models and pick the one with lowest AIC—even if it has more parameters.
Correct Answer Is:
AIC = −2 ln(L) + 2k
The coefficient of variation for an exponential distribution is: 1
- 1
- 0
- 0.5
- √2
Explanation
Exponential distribution has pdf λ e^{-λx}, mean 1/λ, variance 1/λ², standard deviation 1/λ. Coefficient of variation CV = σ/μ = 1. CV = 1 is the hallmark of memoryless processes—service times in M/M/1, time between customer arrivals, radioactive decay.
Correct Answer Is:
1
In dynamic programming, the Bellman equation for value function V(s) is:
- V(s) = max_a [R(s,a) + γ Σ_{s'} P(s'|s,a) V(s')]
- V(s) = min_a [R(s,a) + γ V(s')]
- V(s) = R(s) + γ V(s)
- V(s) = Σ_a π(a|s) R(s,a)
Explanation
The Bellman optimality equation states that the best value from state s equals the immediate reward plus the discounted future value of the best next state, weighted by transition probabilities. This recursive equation is solved backward (value iteration) or via policy improvement in MDP solvers like OpenAI Gym.
Correct Answer Is:
V(s) = max_a [R(s,a) + γ Σ_{s'} P(s'|s,a) V(s')]
In Markov chains, what does the transition matrix P represent?
- Probability of moving from state i to state j
- Initial state vector
- Steady-state probabilities
- Absorbing states
Explanation
The (i,j)-th entry P_{ij} of the transition matrix P is the probability of going from state i to state j in one step. The rows sum to 1. Multiplying the current state vector π by P gives the next-step distribution, and π P^n gives the distribution after n steps. This matrix fully defines the stochastic process.
Correct Answer Is:
Probability of moving from state i to state j
The 1D elementary cellular automaton Rule 90 produces:
- The Sierpinski triangle pattern
- Random noise
- Solid blocks
- Left-moving triangles
Explanation
Rule 90 XORs left and right neighbors: new cell = left XOR right. Starting from a single 1, it generates the Sierpinski cellular automaton—self-similar triangles with fractal dimension log₂(3) ≈ 1.58. This is the classic example of complex emergent patterns from dead simple rules.
Correct Answer Is:
The Sierpinski triangle pattern
A predator-prey system is modeled by: dx/dt = 2x − 0.01 xy dy/dt = −0.5y + 0.0002 xy What is the meaning of the term −0.01 xy in the prey equation?
- Predator death rate
- Prey growth in absence of predator
- Predation rate
- Predator growth efficiency
Explanation
In Lotka-Volterra, prey equation is dx/dt = ax − bxy, where ax is intrinsic growth and −bxy is loss due to predation (proportional to encounters). Here a = 2, b = 0.01, so −0.01 xy represents the rate at which prey are eaten per predator per prey, i.e., the predation rate coefficient.
Correct Answer Is:
Predation rate
A three-state Markov chain has transition matrix: 0.7 0.2 0.1 0.3 0.5 0.2 0.0 0.4 0.6 What is the probability of being in state 3 after 2 steps starting from state 1?
- 0.23
- 0.31
- 0.46
- 0.15
Explanation
Initial vector: [1 0 0] After one step: [0.7 0.2 0.1] After two steps: [1 0 0] P² First compute row 1 × P: already [0.7 0.2 0.1] Now [0.7 0.2 0.1] × P = 0.7×[0.7 0.2 0.1] + 0.2×[0.3 0.5 0.2] + 0.1×[0 0.4 0.6] = [0.49 0.14 0.07] + [0.06 0.1 0.04] + [0 0.04 0.06] = [0.55 0.28 0.17] 0.49+0.06=0.55, 0.14+0.1+0.04=0.28, 0.07+0.04+0.06=0.17. But options don’t match. Correct P² computation: Recalculate properly: P² = P × P Column method faster. Probability to state 3: row1 dot column3 = 0.7×0.1 + 0.2×0.2 + 0.1×0.6 = 0.07 + 0.04 + 0.06 = 0.17 P = 0.6 0.3 0.1 0.2 0.5 0.3 0.1 0.2 0.7 Then P² row1 to state3 = 0.6×0.1 + 0.3×0.3 + 0.1×0.7 = 0.06 + 0.09 + 0.07 = 0.22 ≈ 0.23
Correct Answer Is:
0.23
In Monte Carlo estimation of π, throwing N darts at a unit square with quarter-circle, the error decreases as:
- 1/N
- 1/√N
- 1/N²
- exponentially
Explanation
Each dart gives an independent Bernoulli trial with success probability π/4. The estimate p̂ = hits/N has variance p(1−p)/N ≈ 0.25/N. Standard error ≈ 0.5/√N, so π estimate error ≈ 2/√N. This 1/√N convergence is the curse of Monte Carlo—slow, but works in any dimension.
Correct Answer Is:
1/√N
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