According to wikipedia, and the usage among some math SE results of "equal in distribution" search, to write that 2 random variables $X$ and $Y$ are equal in distribution one writes as $X\overset{d}{=} Y$. I thought based on my lectures and this notes (in page 107), and exercise 4 of this problem set, that this way is also valid: $X \sim Y$. Is this used in some books, or has some acceptance somewhere you can refer to me?
Notation: Is the notation $\sim$ used for "equal in distribution"?
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probability-theory
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3 Answers
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I have never seen people write $ X\sim Y $ for "$X$ and $Y$ have the same distribution". The notation $P^X=P^Y$ is used in Jacod and Protter's Probability Essentials.
On the other hand, in probability theory, the symbol "$\sim$" is usually used in this format: $$ X\sim \textrm{"some distribution"}. $$
For instance, it is usually written that $ {\displaystyle X\sim{\mathcal {N}}(\mu ,\,\sigma ^{2})}, $ where ${\mathcal {N}}(\mu ,\,\sigma ^{2})$ denotes the normal distribution.
Similarly, one writes $X\sim Poi(\lambda)$ for Poisson distribution, and $X\sim B(n,p)$ for binomial distribution.
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0Is it also used for other famous well known distributions, like geometric, binomial, multinomial, poisson? – 2017-01-06
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0Yes, indeed. Normal distribution is an example. – 2017-01-06
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0In statistics, I have seen $X \overset{H_0}{\sim}Y$ for "$X$ and $Y$ have the same distribution under the null hypothesis $H_0$". – 2017-01-06
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In geometry, '$\sim$' has a fairly fixed (standard) meaning, namely "is similar to."
In other fields of math the usage of this symbol varies widely and it doesn't generally have a standardized or a fixed meaning—unless you count that it is usually used to indicate a binary operator of some sort. (A binary operator is one that accepts two operands.)
In fact, I've generally seen $\sim$ used as the symbol of first resort when one wants to define one's own binary operator.
It may also refer to equivalence classes; in other words $A \sim B$ may be commonly used to signify that $A$ and $B$ belong to the same equivalence class.
Actually, the geometric meaning "is similar to" is an instance of this last definition, since similar geometric figures form an equivalence class; that is, any geometric figure is similar to itself, and if a geometric figure is similar to each of two others, those others are also similar to each other.
If I were to use this symbol in a paper and assign it a new meaning, I would probably use it to signify an equivalence class. It could be assigned other meanings, though.
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Although one can understand what this implies, it is not a usual notation for equality in distribution and I have not seen that in a textbook before.
"$\sim$" as in $X\sim \mathcal{N}(\mu, \sigma^2)$ is usually used when a random variable has a "classic" distribution that is well known and can be represented by an abbreviation as well as the corresponding parameters.