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I have never taken probability theory, and I wonder whether one can express some random variable $X$ with mean $\mu$ and variance $\sigma ^2$ in terms of $\mu$ and $\sigma$ only. Or at least something close to it.

Thank you for your help.

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I'd first remark that upper case Roman letters (eg. $X$) are probably a less confusing notation for random variables. Also, I don't quite understand your question; do you want $\mu$ and $\sigma $ to be the sole parameters of $x$'s distribution? –  user17794 Jun 13 '12 at 23:14
    
You cannot express a random variable in terms of its mean and variance, but you can express its density function or distribution function etc in terms of the mean and variance. For example, the density function of a Gaussian (a.k.a. normal) random variable can be expressed in terms of its mean and variance: $f(x) = (2\pi\sigma^2)^{-1/2}\exp((x-\mu)^2/2\sigma^2)$. –  Dilip Sarwate Jun 13 '12 at 23:14
    
@TimDuff if it is possible. Thats my question. Is it possible and if it is not then what is the closest way of doing it? –  Koba Jun 13 '12 at 23:21
    
@DilipSarwate well. I saw somewhere that $$Z=\frac{X−μ}{σ}$$ –  Koba Jun 13 '12 at 23:40
    
Your $Z$ will have mean $0$ and standard deviation $1$, telling you its location and scale but not its shape. It will not have a normal distribution (or any other particular shape of distribution) unless $X$ does. –  Henry Jun 14 '12 at 0:08

2 Answers 2

up vote 3 down vote accepted

No - the mean and variance (if they exist) tell you about the location and scale of a distribution but nothing about its shape.

If a random variable $X$ has mean $\mu$ and variance $\sigma^2$ then the random variable $Y$ defined by $Y=aX+b$ for real $a$ and $b$ has mean $a\mu+b$ and variance $a^2 \sigma^2$.

So given a particular shape of distribution (well behaved enough to have a mean and variance), it is possible to find a distribution which has the same shape but any mean and (positive) variance you specify.

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A Poisson random variable has mean lambda and variance lambda. In the case of one parameter distirbutions the mean and variance will both be functions of a single parameter.

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