DOR:

-1. objective function the summation of bivariate function with a knapsack constraint

develop a DP state, stage, recursion, value function

What if add another term \(x_n x_1\)

-2. what the relation between

\(\min cx s.t. Ax \geq b\) and \(\max cx s.t. Ax \leq b\) when these two LPs have finite optimal solutions.

In fact, they will have the same extreme point to reach the optimal value.

-3. Convex function.

-4. Dual LP and surrogate relaxation.

-5. Lagrangian multiplier to solve \(x_1+ x_1x_2 + x_1x_2x_3 s.t. x_1+x_2+x_3=c\) when n how to extend the property.

SOR:

  1. two-dimensional simple random walk Zn =Xn+Yn Markov chain and simple walk.
\[P_{i,i+1}, P_{i,i-1}\] \[P(Z_n=0)\] \[P((X_30, Y_30)=(1,2))\]
  1. \(M/G/1+ \infty\) waiting space

binomial distribution when given poisson arrivals

the probability of first complete services before second arrives

M(t) indicates the number of customers finish the service by time t

E[M(t)]

\(M/M/1+ \infty\) waiting space + abandonment rate for each waiting customer.

Q matrix

long-run average rate of abandonment rate

  1. (s,S) policy

1） E[T] time between successive orders

2）lim_t\to\infty P(I(t)>x) the probability of inventory is larger than x in [s,S]. I think it should use Wald’s theorem.

3）\(S_n = Y_1+\ldots+Y_n\) \(P(S_n > S)\) convolusion of exponential distribution

4) inventory holding cost

\[S-S_1,S-S,\ldots,S-S_n\]

expected cost in one cycle.

by revewal theorem \(\lim_{t\to\infty} E(t)/t\) = E[cost in one cycle]/E[one cycle time]

ES:

  1. Use median to find a consistent estimator for \theta which is the mean of the expenontial distribution.

median \(T_1 = c \xi\)

Find c.

mean T2, Asymptotic Relative Efficiency of Sample Mean and Median

  1. uniform \([0,\theta]\)

test statistic X_n for \theta.

CR bound by fisher information

\(n(\hat{\theta} - \theta)\) Asymptotic distribution

CI shortest region

  1. xi follows gamma distribution \((2, \theta_i c_i)\).

linear combination of the unbiased estimator.

  1. non parameter statistic how to calculate E(f(x)), Var(f(x))

the bias in fact.

EDF: Empirical Distribution Function

the property of KDE to find the bandwidth.

  1. linear regression with time series.

autoregression structure \(u_i = \rho u_{i-1} + \epsilon_i\)

What is \beta?

Find \(\rho\) by the estimator of \(\hat{u_i}\).