Module 3 · Distributions · Topic 8
Finding Probabilities. To find P(X < value) for a normal distribution: convert to a z-score, then look up the cumulative probability. Most statistics tools and calculators can do this directly.
Finding Values from Probabilities. Sometimes you know the probability and need the value. Use the inverse normal: find the z-score from the probability, then convert back with x = μ + zσ.
P(X = k) = C(n,k)p^k(1-p)^(n-k) — Exact probability of k successes in n independent Bernoulli trials.z = (x - mu) / sigma — Standardizes a value by measuring standard deviations from the mean.Lightbulb life: μ = 1200 hours, σ = 100 hours. P(X < 1050) = P(Z < (1050 − 1200)/100) = P(Z < −1.5) ≈ 0.0668. About 6.7% of bulbs last less than 1050 hours.
What score separates the top 10%? P(Z > z) = 0.10, so P(Z ≤ z) = 0.90. From the z-table, z ≈ 1.28. If μ = 75 and σ = 8, then x = 75 + 1.28(8) = 85.24.
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