Can a discrete random variable be normally distributed?

Question

This question may seem quite strange, but I am wondering whether the above is true?

In several russian books I saw the notion of "somehow" normally distributed discrete r.v. Can this even be possible?

Thank you in advance!

No. Sometimes you can approximate calculations by treating a discrete random variable as a normal random variable. For example, the percentage out of 1000 coin tosses that ended up heads. — Thomas Andrews, May 04 '21 at 06:22
I’ve not heard of such a notion, but clearly it isn’t possible in the literal sense. Are you able to provide any examples where this term is used? There are well-known phenomena that arise in number theory where an integer-valued function can be modeled by a normal distribution, but those are not even random variables, formally. — Erick Wong, May 04 '21 at 06:31
I think the only situation where this might happen is if you consider a degenerate normal distribution. A random variable with such a distribution has zero variance and it is essentially a constant with probability $1$. — StubbornAtom, May 04 '21 at 07:32

score 8 · Accepted Answer · answered May 04 '21 at 07:01

A discrete probability distribution is a probability distribution that can take on a countable number of values.

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A Normal Distribution is a type of continuous probability distribution for a real-valued random variable.

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A continuous probability distribution is a probability distribution whose support is an uncountable set, such as an interval in the real line.

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In mathematics, the support of a real-valued function $f$ is the subset of the domain containing the elements which are not mapped to zero.

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Looking at the definitions of "continuous probability distribution" and "support", we see that the domain of a Normal distribution function must be uncountable, and therefore cannot be countable, which is what is required for it to be a discrete probability distribution, as in the first definition.

score 3 · Answer 2 · answered May 04 '21 at 06:20

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The central limit theorem is applicable to discrete and continuous variables alike. In particular, the binomial distribution tends to a normal one as $n\to\infty$, and so may be called "almost normal".

answered May 04 '21 at 06:20

Parcly Taxel

105,904

But if, for example, when $n$ does not tend to $\infty$ the discrete r.v. can take only 3-5 values ($n < 30$), even though the mean of the data is equal to the median, can we classify such an r.v. as a normally distributed one by CLT? – o.spectrum May 04 '21 at 06:23
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The answer to OP's question is a clear NO. – Kavi Rama Murthy May 04 '21 at 06:27
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@KaviRamaMurthy It depends on what "somehow" is. – Parcly Taxel May 04 '21 at 06:42

score 2 · Answer 3 · answered May 04 '21 at 07:04

The short and strict answer to your question is no. By definition, all normally distributed random variables are continuous random variables. A random variable cannot be continuous and discrete at the same time, so the definition excludes the existence of normally distributed discrete random variables.

The longer answer is "It depends on what 'somehow' means". There is a very famous theorem called the central limit theorem which tells you that random variables that can be written as the average of a large number of independent random variables will be "close" to a normal distribution.

In particular, if $X_i$ are independent variables with mean $\mu$ and variance $\sigma$, then we can define $\overline{X_n}=\frac{1}{n}(X_1+\cdots X_n)$, and the theorem tells us that $\sqrt{n}(\overline{X_n} - \mu)$ converges in distribtution to a normally distributed random variable with mean $0$ and variance $\sigma^2$.

Can a discrete random variable be normally distributed?

3 Answers3

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