Random variables and distributions

Lesson 1 described data that had already been collected; Lesson 2 gave the rules of probability. This chapter joins them. A random variable attaches a number to each outcome of an uncertain process: the number of defective parts in a sample, the number of vehicles arriving at a ramp in a minute, the life of a bearing, the strength of a concrete cylinder. A probability distribution says how likely each value is. A handful of standard distributions model a remarkable range of engineering situations, and knowing which one fits, and why, is a core engineering skill.

Discrete random variables

A discrete random variable takes countable values (0, 1, 2, …). Its probability mass function (pmf) \(p(x) = P(X = x)\) lists the probability of each value; the probabilities sum to 1. The two most important summaries are the mean (expected value) and the variance:

The mean is the long-run average value; the standard deviation \(\sigma\) measures spread, exactly as for data in Lesson 1, but now for the whole population of possible outcomes rather than one sample.

The binomial distribution

Many problems count successes in a fixed number of independent trials, each with the same probability: defective items in a sample, days with rain in a week, components that survive a test. If there are n trials with success probability p, the number of successes X is binomial:

The binomial coefficient counts the arrangements of k successes among n trials (Lesson 2), and \(p^k(1 - p)^{n - k}\) is the probability of any one arrangement.

The Poisson distribution

When events occur randomly and independently in time or space at an average rate λ (per minute, per kilometre, per square metre), the number of events in an interval follows the Poisson distribution:

Vehicle arrivals at light traffic, calls to a service desk, flaws along a pipeline weld, cracks per square metre of pavement, and large earthquakes in a region are all modelled this way. The Poisson distribution is also the limit of the binomial when n is large and p small, with λ = np: for 1000 items at 0.2% defective, both give P(0) = 0.135.

Continuous random variables

A continuous random variable, such as a strength, a time or a length, can take any value in a range. Probability is now an area under a probability density function (pdf) f(x):

The probability of any single exact value is zero; only intervals have probability. The cumulative distribution function \(F(x) = P(X \le x)\) accumulates the area from the left. The mean and variance use integrals in place of sums: \(\mu = \int x f(x)\,dx\) and \(\sigma^2 = \int (x - \mu)^2 f(x)\,dx\).

The exponential distribution and component life

The time between Poisson events, or the life of a component that fails at a constant rate λ (no wear-out, no infant mortality), is exponential:

R(t) is the reliability function. The mean life 1/λ is called the mean time to failure (MTTF). The exponential distribution is memoryless: a component that has survived 1000 hours is as good as new, which is a reasonable model for electronics during their useful life but not for parts that wear, such as bearings and brake pads. Those are usually modelled with the Weibull distribution, which allows a failure rate that rises or falls with age.

The normal distribution

The normal (Gaussian) distribution is the familiar symmetric bell curve with mean μ and standard deviation σ:

It describes measurement errors, manufacturing dimensions, and many material properties, and, through the central limit theorem below, averages of almost anything. About 68% of values lie within one σ of the mean, 95% within two (more precisely 1.96σ) and 99.7% within three.

Every normal distribution becomes the standard normal (mean 0, standard deviation 1) by the z-transformation of Lesson 1:

Φ(z) is the standard normal CDF, tabulated in every statistics text and available as NORM.S.DIST in spreadsheets and scipy.stats.norm.cdf in Python. Useful values: Φ(1.645) = 0.95, Φ(1.96) = 0.975, Φ(2.326) = 0.99.

This is why concrete is batched to a mean strength well above the specified strength f′c. ACI 318, for example, requires a target average strength of at least \(f'_c + 1.34s\) (and \(f'_c + 2.33s - 3.5\) MPa), where s is the plant's documented standard deviation. For f′c = 30 MPa and s = 3.5 MPa both rules give about 34.7 MPa. A plant with tighter control (smaller s) can batch closer to f′c and save cement: statistics has a direct price.

Combining random variables

Engineering quantities are often sums or differences of random parts: the length of a stack of parts, the clearance between a shaft and a hole, a total load. For independent variables:

Variances add, even for a difference. Standard deviations therefore combine by root-sum-square: three parts each with σ = 0.05 mm stack to σ = \(\sqrt{3}(0.05) = 0.087\) mm, not 0.15 mm. This is the statistical tolerancing that ENGG-160 contrasted with worst-case stack-up. Sums and differences of independent normal variables are themselves normal.

The central limit theorem

If you average n independent measurements from any distribution with mean μ and standard deviation σ, the average \(\bar{X}\) has mean μ and standard deviation

and, as n grows, its distribution approaches the normal, whatever the shape of the original distribution. This is the central limit theorem, and it is why the normal distribution is everywhere: many measured quantities are effectively sums of many small independent effects, and statistical methods built on averages work even for skewed data once samples are moderately large (n of about 30 is a common rule of thumb; less for symmetric data).

Computing distributions in Python

Tables are fine for exams, but in practice probabilities come from software. scipy.stats provides every common distribution with the same interface: pmf or pdf, cdf, ppf (the inverse CDF, for percentiles) and sf (the survival function, 1 − cdf).

from scipy import stats

bolts = stats.binom(n=20, p=0.03)
print(f"Binomial P(0) = {bolts.pmf(0):.3f}, P(>=2) = {bolts.sf(1):.3f}")

ramp = stats.poisson(mu=4)
print(f"Poisson P(>6) = {ramp.sf(6):.3f}, 95th percentile = {ramp.ppf(0.95):.0f}")

concrete = stats.norm(loc=40, scale=3.5)
print(f"Normal P(<35) = {concrete.cdf(35):.3f}, 10th percentile = {concrete.ppf(0.10):.1f} MPa")

sensor = stats.expon(scale=2000)
print(f"Exponential R(1000) = {sensor.sf(1000):.3f}")

Output: Binomial P(0) = 0.544, P(>=2) = 0.120; Poisson P(>6) = 0.111, 95th percentile = 8; Normal P(<35) = 0.077, 10th percentile = 35.5 MPa; Exponential R(1000) = 0.607.