Does correlation have to be in the context of (Gaussian) normal distribution?

I am not quite familiar with the concept of correlation. The Pearson's correlation coefficient is defined as:

$\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y}$

Which makes use of Mean and Standard deviation. But, is it strict to the normal distributed data ? Since Gaussian distribution is configured by mean and variance.

I currently have some which is apparently not following normal distribution. When assessing the correlation between them, is correlation appropriate here ?

• The correlation coefficient exists whenever the relevant expectations exist. Very much not confined to the normal. – André Nicolas Jun 4 '14 at 20:14

As a non-normal distribution example, if you have two independent Poisson distributed random variables, $$X$$ with mean $$\lambda$$ and $$Z$$ with mean $$\mu$$, and you let $$Y=X+Z$$ then you can make the following useful statements:
• $$Y$$ is a Poisson distributed random variable with mean $$\lambda + \mu$$
• $$X$$ has standard deviation $$\sigma_X^{\,}=\sqrt{\lambda}$$
• $$Y$$ has standard deviation $$\sigma_Y^{\,}=\sqrt{\mu}$$
• the covariance of $$X$$ and $$Y$$ is $$\mathrm{cov}(X,Y)=\lambda$$
• the correlation of $$X$$ and $$Y$$ is $$\rho_{X,Y}^{\,}=\mathrm{corr}(X,Y)=\dfrac{1}{\sqrt{1+\frac{\mu}{\lambda}}}$$