Normal Distribution: Density, the Bell Curve, the z-Table, the 68–95–99.7 Rule

Why the sum of many dice looks like a bell curve, and how to compute probabilities for a continuous random variable. From a histogram to a density: a probability is an area under the curve, that is, an integral. The normal curve: what μ and σ do, the peak and the inflection points found with the derivative, E(X) = μ and Var(X) = σ². Why the integral of e^(−x²/2) needs a table: the z-table Φ(z) (the area to the left of z), the z-score z = (x − μ)/σ and the probability that X lies between a and b, Φ((b − μ)/σ) − Φ((a − μ)/σ). The 68–95–99.7 rule: 68.27%, 95.45%, 99.73%. Animated on the board: no voice, just the mathematics. 0:00 From a histogram to a density 1:45 The normal curve 3:39 The z-table 5:34 The 68–95–99.7 rule 7:10 Summary #probability #normaldistribution #statistics #math #tutor