Simulation and Randomness

Name: Simulation and Randomness
Author: StemPy Lab

Chance is a part of science, and the computer can imitate it. In this course you make random numbers with a seeded generator, so that a simulation can be repeated, and see the law of large numbers. You estimate probabilities and the number pi by the Monte Carlo method, and see that its error falls as one over the square root of the number of trials. You simulate random walks and find that the typical distance grows as the square root of the number of steps, which is diffusion. You simulate radioactive decay and see why small samples are noisy and large samples are smooth. A project asks how sure a measurement is: the standard error, a whole experiment repeated thousands of times, the bootstrap, and the difference between random and systematic errors. The practice runs your code and tells you which results are wrong.

5 lessons, about 3 h 15 min5 simulations10 problems10 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (5 lessons)

Lesson 1about 40 min
Random numbers with NumPy
A computer makes random numbers with a formula, so they are pseudo-random, and a generator made with np.random.default_rng and a seed gives the same numbers every time, which makes an experiment repeatable. rng.random gives decimals from 0 up to 1, rng.integers gives whole numbers such as the faces of a die, rng.uniform a range, rng.normal numbers from a bell curve with a given centre and spread, rng.choice a pick with optional weights and without replacement, and rng.shuffle changes an order. np.bincount counts the faces. The share of an outcome comes closer to its probability as the number of trials grows, which is the law of large numbers: 600 rolls give shares within about 0.04 of one sixth, and a million rolls within 0.0002.
1 simulation2 practice questions5 Python programs2 problems1 Python lab1 question set
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Lesson 2about 40 min
Monte Carlo: estimating by chance
The Monte Carlo method estimates a quantity by random trials. The probability of an event is estimated as the share of trials in which it happens: for two dice, the sum is 7 or more in 58.4 percent of 200 000 trials, near the exact 21 over 36, and a group of 23 people shares a birthday with a probability of about 0.5. Throwing random points at a square and counting the share inside a quarter circle gives pi over 4, so four times the share estimates pi. The error of a Monte Carlo estimate is about proportional to one over the square root of the number of trials, so a hundred times more trials give an estimate that is ten times more accurate, which is slow, but the method works for problems with no formula. The accuracy can be seen by repeating the whole experiment.
1 simulation2 practice questions4 Python programs2 problems1 Python lab
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Lesson 3about 40 min
Random walks and diffusion
In a random walk on a line, every step is one to the left or one to the right with equal chance, and the position is the running total of the steps, made with np.cumsum. Many walkers are simulated at once as the rows of a two-dimensional array, with cumsum along axis 1. The average position stays at 0, but the walkers spread out: the mean of the squared distance after n steps is about n, so the typical distance is the square root of n and not n. The final positions of many walkers form a bell curve, which is the central limit theorem. This is how particles diffuse: the mean squared distance grows in proportion to the time, and the slope of the straight line of the mean squared distance against the steps is 1 for steps of size 1.
1 simulation2 practice questions5 Python programs2 problems1 Python lab
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Lesson 4about 35 min
Simulating radioactive decay
In radioactive decay, each atom decays at a random moment, with a probability p in each second. A simulation follows the number of atoms that are left: in each second, rng.binomial of the atoms and p gives how many decay. The result follows the expected curve n0 times one minus p to the power t, but it fluctuates, and the fluctuation of a count is about the square root of the count, so a small sample is rough and a large sample is smooth. With p equal to 0.05 the expected half-life is ln 2 over minus ln of 1 minus p, which is 13.5 seconds, and a simulation that counts whole seconds finds 14. The exponential law of decay is not a rule for single atoms, whose decay is unpredictable, but the result of the average over a huge number of atoms.
1 simulation2 practice questions3 Python programs2 problems1 Python lab
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Lesson 5about 40 min
Project: how sure is a measurement?
A measurement has a random error and perhaps a systematic one. Random errors are reduced by repeating: the standard error of the mean is the sample standard deviation with ddof equal to 1 divided by the square root of n, so four times more measurements halve it. A simulated experiment can be repeated thousands of times to see the true spread of the answer: a pendulum experiment that finds g from the slope of T squared against L, with a scatter of 0.01 seconds in each period, gives g equal to 9.81 with a spread of about 0.12. The bootstrap estimates the uncertainty from the data alone, by taking the mean of many samples drawn from it with replacement. A systematic error, such as an instrument that reads 0.3 too high, is not reduced by averaging: the standard error goes to zero but the mean stays 0.3 above the truth.
1 simulation2 practice questions5 Python programs2 problems2 Python labs
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Test Your Learning

Simulation and Randomness: Predict the Output

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Simulation and Randomness

Course outline (5 lessons)

Random numbers with NumPy

Monte Carlo: estimating by chance

Random walks and diffusion

Simulating radioactive decay

Project: how sure is a measurement?

Test Your Learning

Your teacher

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