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If we define a measurable function on a probability space as a function that maps from the sample space to the real numbers, and treat all the people in a Country as the sample space, shouldn't the image of the random variable be a finite set? I don't understand why we would say that the random variable is countable.

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    $\begingroup$ Your objection may perhaps be valid for Vatican City, but the continuous assumption is a way better approximation to reality with any sufficiently populated country. Strictly speaking, the height is also never measured with arbitrary precision (but rather as an integer number of centimeters, say) and even principally the Planck length (as well as the definition of end points) would cause problems. In fact, if you wanted to treat this as a finite set, you'd have to be able to list all these finitely many values before beginning any analysis - which you'd want to do based only on a small sample $\endgroup$ – Hagen von Eitzen Jan 13 at 19:58
  • $\begingroup$ @Hagen von Eitzen Suppose we listed all the people in a country as {$ a_n$}. Then the random variable X would send each $a_n$ to some real number $ b_n$ representing the height. Then we have a countable list of possible heights of people in a country, namely {$ b_n$}. Does this not mean that the image of X is countable, and thus discrete? $\endgroup$ – folo polo Jan 13 at 20:36
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Assuming data are normal is a practical convenience. A discrete random variable that has very many possible values requires a long list of possible values and associated probabilities.

In the US, heights of people are usually rounded to the nearest inch. (A person's height can fluctuate by about half an inch during the day--usually taller in the AM shorter in the PM. So rounding to the nearest inch doesn't lose crucial information.)

Almost all US adults are between 50 and 90 inches tall, so in using a discrete random variable there would be about 40 different values to list along with probabilities of each. Saying that heights are approximately normally distributed with mean 68 and standard deviation 3.5 allows us to get by with 2 parameters.

Theoretically, a normal distribution takes all values on the real line, but an interval $(56,89)$ with $99.9\%$ of the probability is manageably short--even if it theoretically contains uncountably many values. Also one ignores the fact that this normal distribution allows a very tiny probability of (practically impossible) negative heights.

According to this normal distribution, what is the probability that a randomly chosen person is 68in tall? The theoretical answer is $0.$ (if you have uncountably many points you can't assign positive probabilities to any of them.) If you mean $68.00000$ inches tall, then the practical answer is also essentially $0.$ (Who could measure that?) It is more useful to ask for the probability that person is between $67.5$ and $68.5$ (so that the height would be rounded to "68". Then the answer is $0.1136.$

If pilots for a small military airplane need to be shorter than 65 inches, then almost 20% of the population (including lots of women) are still potential candidates as pilots. All such computations are understood to be approximate. Very few, if any, statisticians believe any natural phenomenon is exactly normally distributed.

Sometimes, the gulf between measure theoretic probability and applied statistics can be very wide.


Computations using R software, where pbinom is a continuous normal cumulative distribution function, and qnorm is its inverse (called a quantile function),

qnorm(c(.0005,.9995), 68, 3.5)
[1] 56.48316 79.51684

pnorm(0, 68, 3.5)
[1] 2.212535e-84

diff( pnorm(c(67.5,68.5), 68, 3.5) )
[1] 0.113597

pnorm(65, 68, 3.5)
[1] 0.195683
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