I have two variables 2X and 0.5Y, both are independent and follows Gaussian distribution. How to estimate their mean and variance analytically? I want to know their individual mean and variance, then I want to add them.


closed as unclear what you're asking by Xi'an, Sycorax, Peter Flom Mar 20 at 11:33

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  • $\begingroup$ This question is unclear. It is unclear whether you have some samplea and want to estimate or whether you know variance and mean of X and Y and want to analytically derive variance of Z=2X and W=0.5Y. Also if this is homework remember to tag it accordingly. $\endgroup$ – Jesper Hybel Mar 19 at 9:02
  • $\begingroup$ no its not a homework, I need this for my research work. I have data set name X, which can be distributed as Gaussian with mean µ and variance σ. but in my equation, my variable has a coefficient, like 2X. I need to add 2X with .5Y. If it only X, then I know, the mean and variance is µ and σ respectively. but i am in trouble with this coefficient. I don't know the variance and mean of this data. I want to know how can I mathematically derive mean and variance? How can I add them? $\endgroup$ – Tania islam Mar 19 at 10:38
  • $\begingroup$ Possible duplicate of Sum of normal independent random variables with coefficients $\endgroup$ – Sycorax Mar 19 at 15:04

Let $z = ax$ and $w=by$ with $a$ and $b$ known constants and $x\sim \mathcal N(\mu_x,\sigma_x^2)$ and $y\sim\mathcal N(\mu_y,\sigma_y^2)$ then

$$v = z + w = ax + by$$ is sum of two independent normally distributed variables and hence will be normal.

The mean $\mu_v = \mathbb E[v] = a\mu_x + b \mu_y$ and

The variance $\sigma^2_v = a^2 \sigma_x^2 + b^2\sigma_y^2$ since covariance part is 0 from the assumption of independence.

  • $\begingroup$ yes, thank you so much, But I want to know, How can I know derive mathematically mean and variance by using PDF of gaussian? my mean can be 2µ and variance can be 2σ. How can I know that? $\endgroup$ – Tania islam Mar 19 at 11:06
  • $\begingroup$ Like using Integration or differentiation of PDF of Gaussian $\endgroup$ – Tania islam Mar 19 at 11:12
  • $\begingroup$ Well in that case look up sum of random normals on wiki theres a proof using moment generating function or if you prefer proof by pdf just read the proof there using convolutions. $\endgroup$ – Jesper Hybel Mar 19 at 21:41

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