Normal distribution

What is it about?

For a normally distributed random variable $X \sim N(\mu;\,\sigma^2)$, the probability $P(a \leq X \leq b)$ is the area under the bell curve between $a$ and $b$, and equals $F(b) - F(a)$ with the distribution function $F(x) = P(X \leq x)$; for $N(0;\,1)$ it is called $\Phi$, and $F(x) = \Phi\left(\frac{x - \mu}{\sigma}\right)$. A single value has probability $P(X = a) = 0$.

The bell: symmetric about $\mu$, maximum $\frac{1}{\sigma\sqrt{2\pi}}$ at $\mu$, points of inflection at $\mu \pm \sigma$ (their tangents meet the $x$ axis at $\mu \pm 2\sigma$), total area $1$. Sigma rules: about $68.3\,\%$, $95.4\,\%$ and $99.7\,\%$ within $\mu \pm \sigma$, $\pm 2\sigma$, $\pm 3\sigma$.

Standardising with $z = \frac{x - \mu}{\sigma}$ shifts the bell by $\mu$ and scales it by the factor $\frac{1}{\sigma}$ into $N(0;\,1)$ – the marked area stays the same. Backwards (quantiles): from $P(X \geq x) = 5\,\%$ we get $\Phi(z) = 0.95$, $z \approx 1.645$ and $x = \mu + z\sigma$.

Approximating the binomial: $P(X \leq k) \approx \Phi\left(\frac{k + 0.5 - \mu}{\sigma}\right)$ with $\mu = np$, $\sigma = \sqrt{np(1-p)}$; rule of thumb (Laplace condition): $\sigma > 3$.

Try it

1. Standard normal: drag the bounds and press “μ ± σ”, “μ ± 2σ”, “μ ± 3σ”.
2. Properties of the bell and very small σ.
3. Standardise: press “Standardise” and watch the IQ axis turn into the $z$ axis.
4. Which IQ is exceeded by only 5 %? Press “Find the bound”.
5. Two filling machines and narrow, but shifted.
6. Approximating the binomial and a sample.

Tasks

IQ: $\mu = 100$, $\sigma = 15$. Find $P(X \geq 130)$.

Which IQ is exceeded by only $5\,\%$?

A: $N(500;\,4^2)$, B: $N(502;\,2.5^2)$, tolerance $495$ to $505\,\text{ml}$. Which is better?

100 coin tosses: approximate $P(X \leq 55)$.

Ideas for the classroom

• Let students estimate an area before dragging; the bar in the chart of the distribution function explains the subtraction $F(b) - F(a)$.
• The standardising animation shows why one table for $\Phi$ is enough.
• Common stumbling blocks: reading the height of the curve as a probability; forgetting $1 - F(a)$ for $P(X \geq a)$; looking for $\Phi(z) = 0.05$ instead of $0.95$; forgetting the continuity correction.