Difficulty: Intermediate to Advanced | Prerequisites: Parts 1 and 2 of these study notes, comfort with definite integrals and numerical integration concepts.
The uniform and normal distributions are two of the most widely used probability models. The uniform distribution is the simplest: every outcome in an interval is equally likely. The normal distribution (the bell curve) is the workhorse of statistics, appearing whenever measurements cluster around a central value with symmetric spread. Heights, blood pressures, test scores, rainfall, and brain weights all follow it approximately. Unlike the exponential PDF, the normal PDF has no closed-form antiderivative, so probabilities are computed numerically. The 68-95-99.7 rule and z-scores are the tools that make the normal distribution practical without a calculator for every problem.
The uniform distribution assigns equal probability to every subinterval of equal length. The normal distribution is the symmetric bell curve defined by its mean mu and standard deviation sigma. About 68% of values fall within one sigma of mu, 95% within two, and 99.7% within three. You standardise any normal variable to a z-score using z = (X - mu)/sigma, which converts it to the standard normal (mean 0, standard deviation 1).
Uniform distribution
A distribution where f(x) = 1/(b - a) for a <= x <= b. Every subinterval of a given length has the same probability. Think of it as the "everything is equally likely" distribution.
Normal distribution (Gaussian distribution)
A distribution defined by f(x) = (1/(sigma sqrt(2 pi))) e^(-(x - mu)^2 / (2 sigma^2)). Its graph is the bell curve, symmetric about the mean mu. In simple terms, values cluster near the centre and thin out symmetrically on both sides.
Bell curve
The colloquial name for the graph of the normal distribution, owing to its shape.
Standard normal distribution
The normal distribution with mu = 0 and sigma = 1. Its PDF is f(X) = (1/sqrt(2 pi)) e^(-X^2/2). All normal distributions can be converted to this one using z-scores.
z-value (z-score)
The standardised value z = (X - mu)/sigma. It tells you how many standard deviations X is from the mean. Think of it as a universal coordinate system for normal distributions.
68-95-99.7 rule (empirical rule)
For any normal distribution, approximately 68.3% of values fall within 1 sigma of mu, 95.5% within 2 sigma, and 99.7% within 3 sigma. This is the quick mental shortcut for normal distribution problems.
The PDF is f(x) = 1/(b - a) for a <= x <= b, and 0 elsewhere.
Every subinterval of equal length has the same probability. The probability of X landing in a subinterval of length L is L/(b - a).
The mean is (a + b)/2 (the midpoint of the interval).
Worked example (spinning arrow): An arrow spun around the origin has angle X uniformly distributed on [0, 2 pi). The PDF is f(x) = 1/(2 pi). The probability the arrow points between North and East (0 to pi/2) is (pi/2)/(2 pi) = 1/4.
The normal PDF is f(x) = (1/(sigma sqrt(2 pi))) e^(-(x - mu)^2/(2 sigma^2)).
The mean is mu and the standard deviation is sigma.
The curve is symmetric about mu, so the mean and the median are the same.
The normal PDF has no antiderivative in terms of elementary functions. Probabilities must be computed using numerical integration (a computer, calculator, or z-table).
For any normal distribution:
P(mu - sigma < X < mu + sigma) is approximately 0.6827 (about 68%)
P(mu - 2 sigma < X < mu + 2 sigma) is approximately 0.9545 (about 95%)
P(mu - 3 sigma < X < mu + 3 sigma) is approximately 0.9973 (about 99.7%)
P(mu - 4 sigma < X < mu + 4 sigma) is approximately 0.9999
These percentages hold for every normal distribution, regardless of mu and sigma.
The substitution z = (X - mu)/sigma converts any normal distribution to the standard normal (mu = 0, sigma = 1).
The integral from a to b of (1/(sigma sqrt(2 pi))) e^(-((X - mu)/sigma)^2 / 2) dX equals the integral from alpha to beta of (1/sqrt(2 pi)) e^(-z^2/2) dz, where alpha = (a - mu)/sigma and beta = (b - mu)/sigma.
This means you only need one set of tables (the standard normal table) to handle every normal distribution.
A z-value between -2 and 2 indicates the value falls within two standard deviations of the mean, which covers about 95% of observations.
Blood pressure (Example 12): Systolic blood pressure is normally distributed with mu = 119.7 mm Hg and sigma = 10.9 mm Hg.
P(140 <= X <= 160) is approximately 0.031 (about 3% have Stage 1 hypertension).
P(160 <= X <= 180) is approximately 0.00011 (about 0.011% have Stage 2 hypertension).
P(90 <= X <= 120) is approximately 0.508 (about 51% have normal blood pressure).
ACT scores (Example 13): Scores modelled as normally distributed with mu = 21.1, sigma = 5.1.
P(18 <= X <= 24) is approximately 0.444, so about 44% scored between 18 and 24.
P(1 <= X < 27) is approximately 0.876, so a student scoring 27 is in the top 12%.
To be in the top 8%, a student needs at least 29 (since P(X > 29) is approximately 0.06, which is within the top 8%).
f(x) = \frac{1}{b - a}, \quad a \le x \le bUniform distribution PDF.
f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-(x-\mu)^2 / (2\sigma^2)}Normal distribution PDF.
f(X) = \frac{1}{\sqrt{2\pi}}\, e^{-X^2/2}Standard normal PDF (mu = 0, sigma = 1).
z = \frac{X - \mu}{\sigma}z-score transformation. Converts any normal variable to the standard normal.
\int_a^b \frac{1}{\sigma\sqrt{2\pi}}\, e^{-((X-\mu)/\sigma)^2/2}\, dX = \int_\alpha^\beta \frac{1}{\sqrt{2\pi}}\, e^{-z^2/2}\, dzStandardisation identity, where alpha = (a - mu)/sigma and beta = (b - mu)/sigma.
The uniform distribution models any situation where all outcomes in a range are equally likely: spinning an arrow, selecting a random point on a line segment, or generating random numbers on a computer. The normal distribution models heights, blood pressure, annual rainfall, cholesterol levels, brain weights, manufacturing assembly times, tyre lifetimes, and standardised test scores (SAT, ACT, GRE). Quality control in manufacturing uses the normal distribution to predict defect rates. Medical studies use it to define "normal" ranges for blood pressure and other biomarkers.
Students often assume you can find an antiderivative for the normal PDF. You cannot. There is no closed-form antiderivative in terms of elementary functions. Use numerical integration or z-tables.
Students sometimes think the 68-95-99.7 rule gives exact probabilities. These are approximations. The exact values are 0.6827, 0.9545, and 0.9973.
Students confuse the value of the PDF f(x) at a point with the probability at that point. For continuous variables, f(x) is a density, not a probability. You must integrate to get a probability.
Students forget to standardise before using z-tables. The raw X value must be converted to z = (X - mu)/sigma first.
⚠️ Be able to set up and evaluate probability integrals for the uniform distribution. These are straightforward definite integrals.
⚠️ Know the normal PDF formula and be able to set up the integral for P(a <= X <= b), even if you evaluate it numerically.
⚠️ Memorise the 68-95-99.7 rule. Exam questions frequently ask for probabilities within 1, 2, or 3 standard deviations of the mean.
⚠️ Know how to convert between X-values and z-scores: z = (X - mu)/sigma. This is tested directly.
⚠️ The blood pressure and ACT score examples are classic exam-style problems. Practise setting up the integral with specific mu and sigma values.
⚠️ For the normal distribution, mean = median (because of symmetry). This is a common true/false item.
True or false: For a uniform distribution on [0, 10], the probability of X falling between 3 and 5 is 0.2.
True. P = (5 - 3)/(10 - 0) = 2/10 = 0.2.
Fill in the blank: The normal distribution is symmetric about the ______.
Mean (mu).
True or false: About 95% of values in a normal distribution fall within one standard deviation of the mean.
False. About 68% fall within one standard deviation. About 95% fall within two standard deviations.
Fill in the blank: To standardise a normal variable, compute z = ______.
(X - mu) / sigma.
True or false: For a normal distribution, the mean and the median are always the same.
True (because of symmetry).
Q: An arrow is spun on [0, 2 pi) with uniform distribution. What is the probability it points between South and West (between pi and 3 pi/2)?
A: P = (3 pi/2 - pi)/(2 pi) = (pi/2)/(2 pi) = 1/4.
Q: Systolic blood pressure is normally distributed with mu = 119.7 and sigma = 10.9. Set up the integral for the probability that a person has Stage 1 hypertension (140 to 160 mm Hg).
A: P(140 <= X <= 160) = the integral from 140 to 160 of (1/(10.9 sqrt(2 pi))) e^(-(X - 119.7)^2 / (2 x 10.9^2)) dX. Evaluated numerically, this is approximately 0.031.
Q: ACT scores are normally distributed with mu = 21.1 and sigma = 5.1. What percentage of students scored between 18 and 24?
A: P(18 <= X <= 24) is approximately 0.444, so about 44%.
Q: A student scored 27 on the ACT. What is their approximate percentile?
A: P(1 <= X < 27) is approximately 0.876, so the student is in approximately the 88th percentile (top 12%).
Q: What is the minimum integer ACT score to be in the top 8%?
A: Check P(X > 28) which is approximately 0.086 (more than 8%), then P(X > 29) which is approximately 0.06 (less than 8%). So 29 is the minimum score for the top 8%.
Q: Convert X = 140 to a z-score if mu = 119.7 and sigma = 10.9.
A: z = (140 - 119.7)/10.9 = 20.3/10.9 which is approximately 1.86. This means 140 mm Hg is about 1.86 standard deviations above the mean.
The normal distribution connects to the Central Limit Theorem in statistics: regardless of the underlying distribution, the average of a large sample tends towards a normal distribution. This is why the normal distribution appears so often in practice. The z-score transformation connects to u-substitution in integration (it is a linear substitution). The uniform distribution connects to random number generation in computer science and to the concept of equally likely outcomes from Part 1 of these notes.
Uniform distribution, normal distribution, Gaussian distribution, bell curve, standard normal, z-score, z-value, z-table, 68-95-99.7 rule, empirical rule, standardisation, blood pressure probability, ACT score probability, numerical integration, symmetric distribution, Section 8.8, calculus probability, mu sigma, standard deviation bands