The graph of the normal probability density function is the normal curve. Probabilities are represented by corresponding areas under the curve. Probability areas Location Symmetry Spread **Property** | **Consequence** Total area | The area is 1, so P(X >= k)=1-P(X<k). Symmetry | For c>0, P(X<mu-c)=P(X>mu+c). Location | Changing mu shifts the curve without changing its shape. Spread | A larger sigma2 gives a wider, lower curve; a smaller sigma2 gives a narrower, higher curve. Three-sigma rule P(mu-sigma<X<mu+sigma) 0.683 P(mu-2sigma<X<mu+2sigma) 0.954 P(mu-3sigma<X<mu+3sigma) 0.997 Write X N(mu,sigma2), where sigma>0. The curve is bell-shaped and symmetric about mu; its mean, median and mode are all mu. A single exact value has probability zero.

The graph of the normal probability density function is the normal curve. Probabilities are represented by corresponding areas under the curve.

\(P(\mu-\sigma<X<\mu+\sigma)\approx0.683\)

\(P(\mu-2\sigma<X<\mu+2\sigma)\approx0.954\)

\(P(\mu-3\sigma<X<\mu+3\sigma)\approx0.997\)

Write \(X\sim N(\mu,\sigma^2)\), where \(\sigma>0\). The curve is bell-shaped and symmetric about \(\mu\); its mean, median and mode are all \(\mu\). A single exact value has probability zero.

Continuous Random Variables

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Property	Consequence
Total area	The area is \(1\), so \(P(X\geq k)=1-P(X<k)\).
Symmetry	For \(c>0\), \(P(X<\mu-c)=P(X>\mu+c)\).
Location	Changing \(\mu\) shifts the curve without changing its shape.
Spread	A larger \(\sigma^2\) gives a wider, lower curve; a smaller \(\sigma^2\) gives a narrower, higher curve.

The Normal Curve and Its Properties

The Normal Curve and Its Properties

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