Lecture 4 Monte Carlo Methods and Simulation

Properties for random number:

• Random pattern
• Long period
• Efficiency
• Repeatability
• Portability

3 generators

LCG

$$X_n=(a{X}_{n-1}+c)(\text{mod}m)$$

• m - modulus
• c - increment
• a - multiplier
• $X_0$ - seed (starting value)
• $X_n$ -

2 models:

• if c = 0, LCG form is called as the multiplicative LCG (known as multiplicative model)
• if a = 1, the LCG form is called as the additive LCG (known as the additive model)
• if a is not 1 and c is not 0, then its a mixed model

$$\begin{array}{r}{x}_0=4\\ x_1=(5X_0+7)(mod8)\\ =27mod8\\ =3\\ \\ x_2=(5X_1+7)(mod8)=6\\ x_3=37mod8=5\\ x_4=32mod8=0\\ x_5=7mod8=7\\ x_6=42mod8=2\\ x_7=17mod8=1\\ x_8=12mod8=4\\ x_9=3\\ x_{10}=6\end{array}$$

If change c=2

$x_0=4$ $x_1=22$ $x_2=0$ $x_3=2$ $x_4=4$ $x_5=6$

Random Variate Generator for Exponential Distribution

$$f(x)=\begin{array}{cc}\lambda e^{-\lambda x}& x>=0\\ 0& otherwise\end{array}$$