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Here the Variance Var(x) formula is given

Here as you can see Var(x) formula

$$Var(X)= \frac{4-\pi}{2}\sigma ^2$$

Now the fact is known that

1)Variance general formula is square of standard deviation = σˆ2

2)and standard deviation = σ

but on LHS of the formula Var(x) is given which is Variance and that is equal to σˆ2 and on RHS also in the formula σˆ2 is included

how to differentiate between the two - the var(x) on left side and sigma on right side ,is there any distinction here ? and what does σˆ2 on RHS mean ?

same goes for other formulae in the pic included - what does σ mean there and is it same as general standard deviation ?

we also see in gaussian normal distribution ( bells graph ) - var(x) is σ and expectation E(x) or mean is μ but not in rayleigh , although the notation is used in the rayleigh mean and var(x) formula

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    $\begingroup$ Read the rest of that wikipedia article. The Rayleigh distribution is related to a pair of uncorrelated normal random variables, each with variance $\sigma^2$ and mean zero. So the $\sigma$ is the standard deviation of those random variables, but only a scale parameter in the Rayleigh distribution. The standard deviation of the Rayleigh distributed random variable is $\sigma$ times a constant factor. $\endgroup$
    – Joe
    Aug 27, 2021 at 4:55
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    $\begingroup$ The table linked at imgur is taken without attribution from the English Wikipedia article. The very first line of that table, which is truncated at the linked image, is "Parameters: scale: $\sigma > 0$." Here, $\sigma$ is a parameter of the distribution, not the variance of the distribution. $\endgroup$ Aug 27, 2021 at 4:58
  • $\begingroup$ @Joe could you please elaborate on - pair of uncorrelated normal random variables, each with variance 𝜎2 and mean zero - i dont seem to understand this part , i understood the part where you explained that 𝜎 is a parameter of rayleigh distribution . $\endgroup$ Aug 27, 2021 at 9:43
  • $\begingroup$ @EricTowers 1) so does this mean the general variance we use the formula in elementary probability and statistics ( high school probability ) - which has simple definitions of mean mode median variance - all of that is related to gaussian normal distribution ?? (because in gaussian distribution those parameters = to the symbol what they depict ) 2) and in reality variance and expectation ( mean ) have a much general and wider definition for different statistical distribution ? $\endgroup$ Aug 27, 2021 at 9:46
  • $\begingroup$ Distributions are often related to other distributions. As it states in that wikipedia article, if the components of a two dimensional vector are uncorrelated normal random variables with mean zero and the same variance $\sigma^2$, then the magnitude of the vector follows the Rayleigh distribution with scale parameter $\sigma$. $\endgroup$
    – Joe
    Aug 27, 2021 at 11:11

1 Answer 1

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$\sigma^2$ here is not "variance " but only a parameter of the distribution. To calculate the variance and verify the result in your link use the definition

$$\mathbb{V}[X]= \mathbb{E}[X^2]-\mathbb{E}^2[X]$$

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