Is there a difference between a population and a set of realisations of a random variable?

Question

Is there a difference between a population and a set of realisations of a random variable? I heard that a sample of a random variable is a set of iid duplicates of the random variable as a process, or perhaps a sample is a realisation of this process.

Answer 1

A simple example would work best here.

Population: Adults aged 20-65 who suffer from heart disease and are patients of a local hospital.

Random variable: Heart rate (beats per minute) of a randomly selected adult from this population.

The population is a well-defined collection of individuals we are interested in studying. The individuals belonging to this population must specify a specific set of inclusion criteria for them to be considered part of that population.

The random variable is a characteristic which can be measured/observed for any randomly selected individual from the population. The characteristic could be quantitative (e.g., heart rate) or qualitative (e.g., gender).

Usually, we measure/observe the value of the random variable not just for one randomly selected individual from the population, but for an entire collection of such individuals, that is, for a random sample of individuals. As an example, if the population includes 1,000 individuals in total, the random sample may include only 200 individuals.

The observed/measured values of the random variable of interest (e.g., heart rate) for the individuals included in the sample are the realizations of that variable. For example, the values of heart rate for the 200 individuals in our random sample are the realizations of the blood pressure variable: 40, 60, 100, ..., 80, 90 beats per minute.

We start to talk about processes involving random variables when those random variables are measured repeatedly over space and/or time, etc. I won't go into that here - you can read up more on stochastic processes and then come back on this forum with a more pointed question.

Addendum

The population is a set of units. A specific characteristic of that unit is a variable. If the characteristic is measured/observed on a unit that is randomly selected from the population of units, then the characteristic is named a random variable.

A random variable has a probability distribution. For example, if the random variable is continuous (e.g., weight), the probability distribution of the variable tells us how likely it is that its value for a randomly selected unit from the population will fall in any given interval of plausible values.

As stated in Section 14.3 of https://rafalab.github.io/dsbook/random-variables.html, "we can estimate the distribution function for a random variable by using a Monte Carlo simulation to generate many realizations of the random variable".

An example of such simulation would be as follows: Start out with a large population of humans. Randomly select one human from this population and measure his/her weight. Put the human back in the population and randomly select another human from the population and measure his/her weight. Repeat this process a large number of times. The distribution of weights for all the humans thus selected will represent an approximation to the probability distribution of weight. The more humans were selected from the population, the better the approximation.

You can use the R code below to play with a concrete example - copy it and paste it in the window available at https://rextester.com/l/r_online_compiler and click Run it.

In the example, we know upfront that the humans in our target population have weights that are normally distributed with a mean of 70 kg and a standard deviation of 5 kg. This normal distribution describes the distribution of weight values in the population - you can call it population distribution of weight. The probability distribution of weight describes the distribution of weight values for (a very, very large number of) humans that were randomly selected from the population. We approximate this latter distribution by considering only 1000 humans randomly selected from the population.

When you run the example, you will see that the probability distribution of weight will be close to the population distribution of weight. But these two distributions are conceptually distinct, as they come to be in different ways, as described above. If you consider a sufficiently large number of humans in your Monte Carlo simulation, the shapes of the two distributions will essentially become identical. But for smaller number of humans, they will be quite different. So the number of realizations of the random variable drives how good your Monte Carlo approximation of the probability distribution of weight is; it also drives how close that approximation is to the population distribution of weight.

The shape of the probability distribution of weight depends on the shape of the population distribution of weight. The latter can be normal or non-normal. But the two distributions would have identical shapes (even though conceptually they have different interpretations because they come to be in different ways).

The plot below shows the population distribution of weight (which is the same in shape as the probability distribution of weight) using a blue line. The approximation to this distribution obtained by Monte Carlo simulations is shown in dotted red colour. Note how the approximation improves as we move from 100 to 10,000 to 100,000 randomly sampled values of weight in our simulations.

R Code

mean <- 70
sd <- 5


xres <- NULL

for (i in 1:1000) {
   
   x <- rnorm(1, mean, sd)
   xres <- c(xres, x)
}
          
hist(xres)
plot(density(xres))

mean(xres)
sd(xres)