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I'm working with a software library that generates random values from the standard normal distribution (mean=0, standard deviation=1).

Suppose the random values represent heights which have a mean=175 and standard deviation=10. They must be positive.

I convert from standard normal to normal using the formula Z=(X-mu)/sigma i.e. X=(Z*sigma)+mu e.g. 173.75=(-0.125*10)+175. In theory it's possible to get a value so small that it would imply a negative height e.g. -25=(-20*10)+175. What's the best way of handling this scenario? Which of the following is most appropriate?

  1. Do nothing - it's so unlikely that it's not worth worrying about
  2. Check for negative values and disregard them, regenerate a new random height
  3. Define an absolute within the Normal Distribution model itself (I don't know if this is theoretically possible).
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1 Answer 1

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If $X \sim N(175, 10^2)$, then $P(X \le 0) = \Phi(-175/10) \le 7.2 \times 10^{-69}$. It is extremely unlikely to get a negative height, but still possible.

What you should do in the end really depends on your application and the reasons for you are simulating from this distribution in the first place. You are right that the actual distribution of heights cannot be normally distributed, but it can be a good approximation.

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  • $\begingroup$ Thanks for computing the exact probability. Since the goal is to run simulations in volumes I'll wrap the call to the random variable generator and if it falls outside an acceptable range then regenerate. $\endgroup$
    – Kunal
    Sep 21, 2019 at 22:55

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