matlab_MonteCarlo_Simulation

Roll of an Unfair Die
Assume the die is biased according to the following empirical distribution:
Roll Value 1 2 3 4 5 6 P(roll) 0.1 0.1 0.2 0.3 0.2 0.1
Stop
Volume Under a Surface
Consider finding part of the volume of the sphere
x y z 1
2 2 2
that lies in the first octant（卦限）,
x 0, y 0, z 0
We use a similar method to calculate the volume
Monte Carlo Roll of an Unfair Die Algorithm
Step 5: p=counter/n
Area Under a Curve (曲边梯形的面积)
Use Monte Carlo simulation to model a deterministic behavior, the area under a curve. Find an approximate value of the area.
Input: Total number of n of random points to be generated in the simulation. Output: VOLUME=approximate volume enclosed under the specified surface z=f(x,y) in the first octant x>0,y>0, z>0
Step 5: calculate P(head)=counter/n Step 6: output Probability of heads, P(head)
Stop
Roll of a Fair Die
Use {1,2,3,4,5,6} to stand for the 6 events when we roll a fair die.
Area Under a Curve (曲边梯形的面积)
Select a point P(x,y) at random within the rectangular region:
[a, b] [0, M ]
How can we do this on a computer? To generate a random number pair (x,y) uniformly distributed (均匀分布)in the region.
0 f ( x) M ,
a xb
Monte Carlo Area Algorithm
Sr i=1,2,…,n, do step 3-5
step 3: calculate random coordinate
xi and yi
Note: the number of simulation should be very large
Monte Carlo Area Algorithm
Input: Total number of n of random points to be generated in the simulation. Output: AREA=approximate area under the specified curve y=f(x) over [a,b], where
Monte Carlo Simulation
中国海洋大学数学学院 施心慧
Contents
Idea of Monte Carlo Simulation Simulating Deterministic Behavior:
• Area Under a Curve(曲边梯形的面积) • Volume Under a Surface(曲顶柱体的体积)
Simulating Probabilistic Behavior:
• Flip（抛、掷） of a Fair（均匀的） Coin • Roll of a Fair Die（骰子） • Roll of an Unfair Die
Idea of Monte Carlo Simulation
Simulate behavior or event by generating random numbers of some distributions
Monte Carlo Roll of an Unfair Die Algorithm
Step 1: Initialize: counter(i)=0,i=1,2,…,6
Monte Carlo Roll of an Unfair Die Algorithm
Step 2: for i=1,2,…,n, do steps 3-4
step 3: obtain a random number satisfying: 0 xi 1 step 4: if
0 xi 0.1, counter (1) counter (1) 1; 0.1 xi 0.2, counter (2) counter (2) 1; 0.2 xi 0.4, counter (3) counter (3) 1; 0.9 xi 1, counter (6) counter (6) 1;
Roll of an Unfair Die
Use random numbers to simulate the events
Roll Value 1 2 3 4 5 6 P(roll) 0.1 0.1 0.2 0.3 0.2 0.1 Value of xi [0,0.1] (0.1,0.2] (0.2,0.4] (0.4,0.7] (0.7,0.9] (0.9,1] side 1 2 3 4 5 6
Area Under a Curve (曲边梯形的面积)
If this procedure is repeated for extremely many times, then we get
area under curve area of rectangle number of points counted below curve total number of random points
Volume Under a Surface
volume under surface volume of box

number of points counted below surface in 1st octant total number of random points
Monte Carlo Volume Algorithm
P(a particular number)
number of occurences of the particular number {1,2,3,4,5,6} total number of trials
Let’s consider a probability model in which each event is not equal likely. Use {1,2,3,4,5,6} to stand for the 6 events when we roll a unfair die.
Roll of an Unfair Die
Monte Carlo Roll of an Unfair Die Algorithm
Input: Total number of n of random rolls of an unfair die in the simulation. Output: Probability for rolls {1,2,3,4,5,6}
Axioms of Probability
Axioms of Probability
One of the keys to understand Monte Carlo simulation is the axioms of probability.
The probability of some event can be considered as the limit of the frequency of that event as the number of trails approaches to infinity.
a xi b, ci yi di , 0 zi M
Monte Carlo Volume Algorithm
step 4: calculate Step 5: if
f ( xi , yi )
zi f ( xi , yi )
Then counter:=counter+1
Step 7: output VOLUME Step 6: VOLUME=M(d-c)(b-a)*counter/n
Monte Carlo Volume Algorithm
Step 1: Initialize: counter=0 Step 2: for i=1,2,…,n, do steps 3-5
step 3: calculate random coordinate
xi, yi and zi
satisfying
Input: Total number of n of random flips of a fair coin to be generated in the simulation. Output: Probability of getting a head when we flip(抛、掷) a fair coin