Sampling from distributions

From uniform to anything

A generator gives numbers spread evenly between 0 and 1. A model needs other shapes: a coin that wins 30% of the time, a waiting time between trades, a normal daily return. The trick is the same: apply a transformation to uniform numbers. The most general is the inverse transform: if F is the cumulative distribution function of what you want, then F⁻¹(u) for uniform u has exactly that distribution. Whole arrays go through the transformation at once.

import math
import numpy as np

# Exponential waiting times with rate 2 (a mean of 0.5): F(x) = 1 - exp(-rate * x), so x = -ln(1 - u) / rate
u = np.array([0.1, 0.5, 0.9])
print((-np.log(1 - u) / 2).round(4).tolist())       # [0.0527, 0.3466, 1.1513]   small u, short wait

rng = np.random.default_rng(42)
waits = -np.log(1 - rng.random(100000)) / 2
print(round(waits.mean(), 2))                        # 0.5   the mean is 1 / rate

Discrete outcomes and the normal distribution

For a discrete table of outcomes, rng.choice(values, size=n, p=probs) does the inverse transform for you: it draws each value with its own probability, such as a day that is -2%, 0% or +3% with probabilities 20%, 50% and 30%. For the normal distribution the inverse of its distribution function has no simple formula, so a classic trick is the Box-Muller transform: from two independent uniform numbers u1 and u2, z = sqrt(-2 ln u1) * cos(2 pi u2) is a standard normal draw. In practice rng.standard_normal(n) does this faster, but writing it once shows there is no magic.

rng = np.random.default_rng(3)
days = rng.choice(np.array([-0.02, 0.0, 0.03]), size=100000, p=[0.2, 0.5, 0.3])
print(round(days.mean(), 3), round(0.2 * -0.02 + 0.3 * 0.03, 3))     # 0.005 0.005   the sample mean matches the table

rng = np.random.default_rng(42)
u1, u2 = 1 - rng.random(100000), rng.random(100000)                   # 1 - random() is in (0, 1]: never log(0)
z = np.sqrt(-2 * np.log(u1)) * np.cos(2 * math.pi * u2)
print(abs(z.mean()) < 0.02, round(z.std(), 2))                        # True 1.0   mean 0 and standard deviation 1

From standard normal draws to prices

A standard normal draw z is a shock; the model decides how it moves a price. The usual model for a stock, geometric Brownian motion, says the price after t years is S = S0 * exp((mu - 0.5 sigma²) t + sigma sqrt(t) z). It is lognormal: always positive, with the long right tail real prices have. The -0.5 sigma² term corrects the average so that the expected price grows exactly at mu: E[S] = S0 * exp(mu t). The median is lower than the mean, which is why a typical outcome is worse than the average one.

s0, mu, sigma, t = 100, 0.08, 0.2, 1
z = np.array([-1.0, 0.0, 1.0])                         # a one-sigma loss, no shock, a one-sigma gain
prices = s0 * np.exp((mu - 0.5 * sigma ** 2) * t + sigma * math.sqrt(t) * z)
print(prices.round(2).tolist())                         # [86.94, 106.18, 129.69]   all positive, the upside is larger

draws = np.random.default_rng(0).standard_normal(100000)
simulated = s0 * np.exp((mu - 0.5 * sigma ** 2) * t + sigma * math.sqrt(t) * draws)
print(round(simulated.mean(), 1), round(s0 * math.exp(mu * t), 1))     # 108.3 108.3   the mean grows at mu
print(round(float(np.median(simulated)), 1))                           # 106.1         the median is lower

# NumPy ships ready-made samplers too; the transformations above show what they do inside
z = np.random.default_rng(0).standard_normal(100000)
print(abs(z.mean()) < 0.02, round(z.std(), 2))      # True 1.0   standard_normal: mean 0, standard deviation 1

• Inverse transform: apply the inverse distribution function to uniform numbers; exponential: -ln(1 - u) / rate.
• rng.choice(values, size=n, p=probs) samples a discrete table; Box-Muller sqrt(-2 ln u1) * cos(2 pi u2) makes normal draws.
• Price after t years: S0 * exp((mu - 0.5 sigma²) t + sigma sqrt(t) z), lognormal; always check a big sample against the exact mean.

Next up: Monte Carlo estimation, using random draws to compute averages and probabilities that have no formula.

Lab · what's graded

• Draw from a table of outcomes with sample_discrete25 pts
• Invert the distribution function with exponential_from_uniform25 pts
• Make bell-curve numbers with box_muller25 pts
• Turn shocks into future prices with terminal_prices25 pts