Probability and Statistics · Continuous distributions

• Uniform(a, b): every value in [a, b] is equally likely.
• Normal(μ, σ): the bell curve. The z-score z = (x − μ)/σ says how many standard deviations a value sits from the mean. The 68–95–99.7 rule gives the share within 1, 2 and 3 σ.
• Exponential(λ): the waiting time between Poisson events. It’s memoryless.

For any continuous distribution:

• cdf(x) = P(X ≤ x), the area to the left
• ppf(q) is the inverse of the cdf, giving the value with area q to its left (quantiles, cut-offs)

from scipy import stats

z = stats.norm(0, 1)
for k in (1, 2, 3):
    print(f"within {k} sd: {z.cdf(k) - z.cdf(-k):.4f}")

h = stats.norm(loc=170, scale=8)      # heights in cm
print("P(height > 185) =", round(h.sf(185), 4), "| z =", (185 - 170) / 8)
print("95th percentile =", round(h.ppf(0.95), 2), "cm")

wait = stats.expon(scale=1 / 4)       # 4 arrivals per hour -> mean wait 15 min
print("mean wait (h)", wait.mean(), "| P(wait > 30 min) =", round(wait.sf(0.5), 4))
print("memoryless:", round(wait.sf(0.75) / wait.sf(0.25), 4), "==", round(wait.sf(0.5), 4))

Output:

within 1 sd: 0.6827
within 2 sd: 0.9545
within 3 sd: 0.9973
P(height > 185) = 0.0304 | z = 1.875
95th percentile = 183.16 cm
mean wait (h) 0.25 | P(wait > 30 min) = 0.1353
memoryless: 0.1353 == 0.1353