Ch. 20.7 Monte Carlo Method

Estimation of \( \pi \) using the Monte-Carlo Method:

Generating Random Points and Estimating \( \pi \)

inside = sum(u(-1, 1, n)**2 + u(-1, 1, n)**2 + u(-1, 1, n)**2 <= 1)

This line performs:

• u(-1, 1, n) generates n random numbers uniformly distributed between -1 and 1. The expression u(-1, 1, n)**2 computes the square of each of these random numbers.
• The process is repeated three times to generate three sets of random numbers, which represent the coordinates \( (x, y, z) \) in a 3D space.
• The code calculates the condition:
$$ x^2 + y^2 + z^2 \leq 1 $$

This condition checks whether each of the generated points lies inside a unit sphere (which is defined by the equation \( x^2 + y^2 + z^2 \leq 1 \)).

• sum(...) counts how many of these points fall inside the unit sphere. The result is stored in the variable inside.

Estimating \( \pi \):

pi = 6.0 * inside / n

estimation of \( \pi \) using the formula:

The volume of the unit sphere is \( \frac{4}{3}\pi \cdot1^3 \), while the volume of the cube defined by the range \( [-1, 1] \) is \( 2^3 = 8 \).

$$ \frac{V_{sphere}}{V_{cube}} = \frac{\frac{4}{3}\pi}{8}$$

$$ \pi \approx \frac{6 \cdot (\text{number of points inside the sphere})}{\text{total number of points}} $$

\[ \int_{a}^{b} x^2 \sin(x/2) \, dx \] using MC method

\[ \int_{a}^{b} x^3 \sin(x) \, dx \] using MC method

\[ \int_{a}^{b} \exp(-2 \cdot |x - 5|) \, dx \] using MCI Importance sampling method