Presented by: Zhenhuan Sui Nov. 30 th, 2009

Stochastic: having a random variable Stochastic process(random process):  counterpart to a deterministic process.  some uncertainties in its future evolution described by probability distributions.  even if the initial condition is known, the process still has many possibilities(some may be more probable) Mathematical Expression: For a probability space, a stochastic process with state space X is a collection of X-valued random variables indexed by a set time T where each F t is an X-valued random variable.

Stochastic model: tool for estimating probability distributions of potential outcomes allowing for random variation in one or more inputs over time random variation is from fluctuations gained from historical data Distributions of potential outcomes are from a large number of simulations Markov property

Andrey Markov: Russian mathematician Definition of the property: the conditional probability distribution of future states only depends upon the present state and a fixed number of past states(conditionally independent of past states) Mathematical Expression: X(t): state at time t, t > 0; x(s): history of states, time s < t probability of state y at time t+h, when having the particular state x(t) at time t probability of y when at all previous times before t. future state is independent of its past states.

Examples: Population: town vs. one family Gambler’s ruin problem Poisson process: the arrival of customers, the number of raindrops falling over an area Queuing process: McDonald's vs. Wendy’s Prey-predator model Applications: Physics: Brownian motion: random movement of particles in a fluid(liquid or gas) Monte Carlo Method Weather Forecasting Astrophysics Population Theory Decision Making

Law of Total Probability Conditional Probability Bayes Theorem Useful Formulas:

Model: Set of strategies: A ={A 1,A 2,…,A m } Set of states: S={S 1,S 2,…,S n }, and its Probability distribution is P{S j }=p j Function of decision-making: v ij =V(A i,S j ), which is the gain (or loss) at state S j taking strategy A i Set of the consulting results: I={I 1,I 2,…,I l }, the quality of consulting is P(I k |S j )=p kj, cost of consulting: C

Max gain before consulting By Law Of Total Probability and Bayes Theorem Max expected gain when the result of consulting is I k Expected gain after consulting YES!NO!

There are A 1, A 2 and A 3 three strategies to produce some certain product. There are two states of demanding, High S 1, Low S 2. P(S 1 )=0.6, P(S 2 )=0.4. Results for the strategies are as below (in dollars): States S1S1 S2S2 A1A1 A2A2 A3 180, , , , , ,000 Results Strategies If conducting survey to the market, promising report: P(I 1 )=0.58 Not promising report: P(I 2 )=0.42 Abilities to conduct the survey: P(I 1 |S 1 )=0.7, P(I 2 |S 2 )=0.6 Cost of consulting and surveying is 5000 dollars. Should the company go for consulting?

v 11 =180000, v 12 = , v 21 = v 22 =-50000, v 31 =100000, v 32 = Expected gain of the strategies: E(A 1 )=0.6× ×( － )=48000 E(A 2 )=0.6× ×( － 50000)=52000 E(A 3 )=0.6× ×( － 10000)=56000 q 11 =P(S 1 |I 1 )=0.72, q 21 =P(S 2 |I 1 )=0.28, q 12 =P(S 1 |I 2 )=0.43, q 22 =P(S 2 |I 2 )=0.57 Result is I 1, max expected gain is Result is I 2, max expected gain is Expected gain after consulting: ER–E(A3)=67202–56000=11202>C=5000YES!!!

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