mathematics//probability//random variable

A random variable is a quantity whose value is not known exactly, together with a description of which values are more credible, and it is how every uncertain number in a system (a sensor reading, a failure time, a delay) enters a calculation. The temperature of a bearing ten minutes from now is one; so is what a drone's gyroscope reads while the drone sits still on the ground. Nothing in the world has to be random in a deep sense: the bearing will have one temperature, and the variable describes what the engineer can say about it beforehand.

A random variable is a quantity whose value is not known exactly, together with a description of which values are more credible, and it is how every uncertain number in a system (a sensor reading, a failure time, a delay) enters a calculation. The temperature of a bearing ten minutes from now is one; so is what a drone's gyroscope reads while the drone sits still on the ground. Nothing in the world has to be random in a deep sense: the bearing will have one temperature, and the variable describes what the engineer can say about it beforehand.

The description is its probability distribution, and the number that actually turns up when the quantity is observed is a realization. Keeping the two apart clears most confusions. A sensor reading of 20.3 °C is a realization; the reading before it arrives is a random variable, centred on the true temperature plus the sensor's bias and spread by its noise. An average of 25 readings is again a random variable, with a narrower distribution, and the one number it produced today is one realization of it.

Functions of random variables are random variables, and their distributions follow.

The angle a drone computes by integrating its gyroscope, the position a filter estimates, the margin left by a stack of machined parts: each is a function of uncertain inputs, so each carries a distribution that can be computed or simulated, and that distribution, more than its central value, is what a safety case is built on.

Discrete variables take values from a list (the state of a valve, the number of failures in a month) and have probabilities; continuous ones take values on a line (a voltage, a time) and have densities. Most physical measurements are continuous and become discrete once an ADC rounds them.

A variable is summarized by its expected value and its variance; two variables, by their covariance, and by whether they are independent, which is a much stronger statement than zero covariance.

A sequence of random variables indexed by time, where each one depends on those before it, is a stochastic process: the noise of a sensor over a whole flight is a process, the noise of one sample is a variable.

Turning realizations back into a statement about the variable (its mean, its spread, its shape) is the work of statistics, and the number it returns, an estimate, is itself the realization of a random variable.