Normal Distribution

The normal distribution is the most important continuous distribution in H2 Maths. Its symmetric, bell-shaped curve is fully described by two parameters — the mean $\mu$ and the variance $\sigma^2$ — and virtually every probability question reduces to either reading off a GC value or working backwards from one.

What you need to know

• Notation: $X \sim \text{N}(\mu, \sigma^2)$. The curve is symmetric about $\mu$; roughly 68 % of values fall within one standard deviation of the mean, and roughly 95 % within two.
• Standardising: convert $X$ to the standard normal $Z \sim \text{N}(0,1)$ by

• Finding probabilities: use the GC (CASIO: STAT → DIST → NORM → Ncd; TI-84: normalcdf) to evaluate $P(a < X < b)$ directly, or convert to $Z$ first. Always sketch the curve and shade the region.
• Inverse normal: given $P(X < x) = p$, use InvN (CASIO) or invNorm (TI-84) to find $x$. Make sure you supply the correct tail probability — the GC expects a cumulative probability from the left.
• Linear combinations — expectation: for independent $X \sim \text{N}(\mu_X, \sigma_X^2)$ and $Y \sim \text{N}(\mu_Y, \sigma_Y^2)$,

• Linear combinations — variance: variances always add, regardless of the sign between the variables:

Worked example 1 — Standardising to find a probability

The mass of a loaf of bread produced by a bakery is modelled by $X \sim \text{N}(800, 625)$, where masses are in grams.

Find the probability that a randomly chosen loaf has mass between 775 g and 830 g.

Solution.

Here $\mu = 800$ and $\sigma = \sqrt{625} = 25$.

Standardise both bounds:

So the required probability is $P(-1 < Z < 1.2)$.

Using the GC (normalcdf with $\mu = 0$, $\sigma = 1$, lower $= -1$, upper $= 1.2$):

Worked example 2 — Inverse normal

The time $T$ (in minutes) a student spends on a particular exam question is modelled by $T \sim \text{N}(12, 4)$.

The teacher wants to identify the slowest 10 % of students. Find the threshold time $t_0$ such that $P(T > t_0) = 0.10$.

Solution.

$P(T > t_0) = 0.10$ means $P(T \leq t_0) = 0.90$.

Using InvN on the GC with area $= 0.90$, $\mu = 12$, $\sigma = \sqrt{4} = 2$:

Students who take longer than approximately 14.6 minutes fall in the slowest 10 %.

Worked example 3 — Linear combination of independent normal variables

The masses of apples and oranges in a market are independently normally distributed:

where masses are in grams.

Find the probability that the total mass of 2 randomly chosen apples and 1 randomly chosen orange exceeds 560 g.

Solution.

Let $W = A_1 + A_2 + B$, where $A_1$ and $A_2$ are independent observations of $A$.

Mean:

Variance (variances of independent variables add):

So $W \sim \text{N}(500, 425)$.

Using the GC (normalcdf, lower $= 560$, upper $= 10^{99}$, $\mu = 500$, $\sigma = \sqrt{425}$):

Common mistakes

• Subtracting variances for a difference of variables. $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$ — variances always add when variables are independent, even if the combination involves a minus sign.
• Using $\sigma^2$ instead of $\sigma$ when standardising. The denominator in $Z = \tfrac{X - \mu}{\sigma}$ is the standard deviation, not the variance.
• Supplying the wrong tail probability to InvN. The GC’s inverse normal function expects the cumulative probability from the left; if the question gives a right-tail probability, subtract from 1 first.
• Forgetting to check independence before combining. The variance-addition formula requires the variables to be independent; if the question does not state independence, you cannot assume it.