The code

Straight from labs/course-4-stats/python/03_random_variables_distributions.py, unchanged.

"""Course 4 Lab, experiments 8-11: random variables, expectation and variance,
and the discrete and continuous probability distributions.

Excel equivalents are named against each section, since the prescribed lab is
a spreadsheet lab (see ../excel-walkthroughs.md).
"""

import random
import statlib as S

random.seed(42)          # reproducible output

# Step 1: Experiment 8: a die rolled 1000 times
print("=" * 66)
print("EXPERIMENT 8: Simulating discrete and continuous random variables")
print("=" * 66)

print("\nDISCRETE -- rolling a fair die 1000 times")
rolls = [random.randint(1, 6) for _ in range(1000)]
print(f"{'Face':<8}{'Observed':<12}{'Expected':<12}Frequency")
for face in range(1, 7):
    observed = rolls.count(face)
    print(f"{face:<8}{observed:<12}{1000/6:<12.1f}{'#' * (observed // 10)}")
print(f"  Excel: =RANDBETWEEN(1,6), then COUNTIF to tally")

# Step 2: 1000 draws from a normal distribution
print("\nCONTINUOUS -- 1000 draws from Normal(mean 100, sd 15)")
sample = [random.gauss(100, 15) for _ in range(1000)]
sample_mean = sum(sample) / len(sample)
sample_var = sum((x - sample_mean) ** 2 for x in sample) / (len(sample) - 1)
print(f"  sample mean = {sample_mean:.2f}   (population mean 100)")
print(f"  sample sd   = {sample_var ** 0.5:.2f}   (population sd 15)")
print("  Excel: =NORM.INV(RAND(), 100, 15)")

# Step 3: Experiment 9: expectation and variance
print("\n" + "=" * 66)
print("EXPERIMENT 9: Expectation and variance from a probability distribution")
print("=" * 66)

# A discrete random variable given by its PMF.
values = [0, 1, 2, 3, 4]
probs = [0.10, 0.25, 0.30, 0.25, 0.10]

print(f"\n{'x':<8}{'P(x)':<10}{'x.P(x)':<12}{'x^2.P(x)':<12}")
print("-" * 42)
expectation = 0.0
second_moment = 0.0
for x, p in zip(values, probs):
    expectation += x * p
    second_moment += x * x * p
    print(f"{x:<8}{p:<10.2f}{x * p:<12.3f}{x * x * p:<12.3f}")
print("-" * 42)
print(f"{'Sum':<8}{sum(probs):<10.2f}{expectation:<12.3f}{second_moment:<12.3f}")

variance = second_moment - expectation ** 2
print(f"\n  E(X)   = sum of x.P(x)          = {expectation:.3f}")
print(f"  E(X^2) = sum of x^2.P(x)        = {second_moment:.3f}")
print(f"  Var(X) = E(X^2) - [E(X)]^2      = {second_moment:.3f} - "
      f"{expectation:.3f}^2 = {variance:.3f}")
print(f"  SD(X)  = sqrt(Var(X))           = {variance ** 0.5:.3f}")
print("\n  The shortcut Var(X) = E(X^2) - [E(X)]^2 is quicker in an exam than")
print("  computing E[(X - mu)^2] term by term, and gives the same answer.")
print("  Check that the probabilities sum to 1 first -- if they do not, the")
print("  question is misread or mistyped.")

# Step 4: Experiment 10: the binomial
print("\n" + "=" * 66)
print("EXPERIMENT 10: Discrete distributions -- Binomial and Poisson")
print("=" * 66)

n, p = 10, 0.3
print(f"\nBINOMIAL(n={n}, p={p})")
print("  Use when: a fixed number of independent trials, each success or")
print("  failure, with a constant probability of success.")
print(f"\n{'k':<6}{'P(X=k)':<12}{'P(X<=k)':<12}Distribution")
for k in range(n + 1):
    pmf = S.binomial_pmf(k, n, p)
    cdf = S.binomial_cdf(k, n, p)
    print(f"{k:<6}{pmf:<12.5f}{cdf:<12.5f}{'#' * int(pmf * 150)}")
print(f"\n  Mean     = n.p        = {n} x {p} = {n * p:.2f}")
print(f"  Variance = n.p.(1-p)  = {n} x {p} x {1-p} = {n * p * (1 - p):.2f}")
print(f"  Excel: =BINOM.DIST(k, {n}, {p}, FALSE) for the PMF, TRUE for the CDF")

lam = 3
# Step 5: The Poisson
print(f"\nPOISSON(lambda={lam})")
print("  Use when: counting events in a fixed interval of time or space,")
print("  occurring independently at a constant average rate.")
print(f"\n{'k':<6}{'P(X=k)':<12}{'P(X<=k)':<12}Distribution")
for k in range(11):
    pmf = S.poisson_pmf(k, lam)
    cdf = S.poisson_cdf(k, lam)
    print(f"{k:<6}{pmf:<12.5f}{cdf:<12.5f}{'#' * int(pmf * 150)}")
print(f"\n  Mean = Variance = lambda = {lam}")
print("  That mean and variance are equal is the signature of a Poisson.")
print(f"  Excel: =POISSON.DIST(k, {lam}, FALSE)")

print("\n  POISSON AS A LIMIT OF THE BINOMIAL (n large, p small, np = lambda):")
print(f"  {'k':<6}{'Binomial(1000, 0.003)':<24}{'Poisson(3)':<14}difference")
for k in range(6):
    b = S.binomial_pmf(k, 1000, 0.003)
    po = S.poisson_pmf(k, 3)
    print(f"  {k:<6}{b:<24.6f}{po:<14.6f}{abs(b - po):.6f}")

# Step 6: Experiment 11: the normal
print("\n" + "=" * 66)
print("EXPERIMENT 11: Continuous distributions -- Normal and Exponential")
print("=" * 66)

mu, sigma = 100, 15
print(f"\nNORMAL(mu={mu}, sigma={sigma})  -- IQ scores, a standard example")
print("\n  THE EMPIRICAL RULE (68-95-99.7)")
for k in (1, 2, 3):
    low, high = mu - k * sigma, mu + k * sigma
    prob = S.normal_cdf(high, mu, sigma) - S.normal_cdf(low, mu, sigma)
    print(f"    within {k} sd  [{low:6.1f}, {high:6.1f}] -> {prob * 100:6.2f}%")

print("\n  SPECIFIC PROBABILITIES")
print(f"    P(X <= 115)      = {S.normal_cdf(115, mu, sigma):.4f}"
      f"   Excel =NORM.DIST(115,100,15,TRUE)")
print(f"    P(X > 130)       = {1 - S.normal_cdf(130, mu, sigma):.4f}"
      f"   Excel =1-NORM.DIST(130,100,15,TRUE)")
print(f"    P(85 <= X <= 115) = "
      f"{S.normal_cdf(115, mu, sigma) - S.normal_cdf(85, mu, sigma):.4f}")

print("\n  STANDARDISING -- z = (x - mu) / sigma")
for x in (85, 100, 115, 130):
    z = (x - mu) / sigma
    print(f"    x = {x:3d}  ->  z = {z:+.2f}  ->  P(X <= x) = "
          f"{S.normal_cdf(z):.4f}")

print("\n  PERCENTILES (inverse -- Excel's NORM.INV)")
for pct in (0.90, 0.95, 0.99):
    print(f"    {pct * 100:.0f}th percentile = {S.normal_ppf(pct, mu, sigma):.2f}")

rate = 0.5
# Step 7: The exponential
print(f"\nEXPONENTIAL(lambda={rate})  -- waiting time until the next event")
print(f"  Mean = 1/lambda = {1 / rate:.2f}, Variance = 1/lambda^2 = "
      f"{1 / rate ** 2:.2f}")
print(f"\n{'x':<8}{'PDF f(x)':<14}{'CDF P(X<=x)':<16}P(X>x)")
for x in (0, 1, 2, 3, 4, 5):
    print(f"{x:<8}{S.exponential_pdf(x, rate):<14.5f}"
          f"{S.exponential_cdf(x, rate):<16.5f}"
          f"{1 - S.exponential_cdf(x, rate):.5f}")
print(f"\n  Excel: =EXPON.DIST(x, {rate}, TRUE) for the CDF")
print("\n  MEMORYLESSNESS -- P(X > s+t | X > s) = P(X > t)")
s, t = 2, 3
joint = 1 - S.exponential_cdf(s + t, rate)
given = 1 - S.exponential_cdf(s, rate)
print(f"    P(X > {s+t} | X > {s}) = {joint:.5f} / {given:.5f} = {joint / given:.5f}")
print(f"    P(X > {t})           = {1 - S.exponential_cdf(t, rate):.5f}")
print("    Equal -- having waited 2 minutes tells you nothing about the next 3.")
print("    The exponential is the only continuous distribution with this property.")

Where this sits

One experiment from the Statistical Foundations for Data Science lab. The rest of them, and the theory behind this one, are on the lab page.