How to find correlation between two random variables? What is the way to find a correlation between two variables, where $X_1,X_2,X_3$ are independent and random variable is a linear equation of these variables. Also, how to find an value of $a_1$, so that two variables are uncorrelated? For example if variables are given as $X_1+2X_2$ and $3X_1+aX_2$. Thanks for any advice!
We have $X_1,X_2$ and $X_3$ independent.

Let $U=a_1X_1+a_2X_2+a_3X_3$ and $V=b_1X_1+b_2X_2+b_3X_3$

$$Corr(U,V)=\frac{cov(U,V)}{\sqrt{var(U)var(V)}}=\frac{\sum_{k=1}^{3}{\sum_{j=1}^{3}{a_kb_jcov(X_k,X_j)}}}{\sqrt{cov(U,U)cov(V,V)}}$$

if $ k \ne j$, $cov(X_k,X_j)=0$

$$Corr(U,V)=\frac{a_1 b_1var(X_1)+a_2 b_2var(X_2)+a_3b_3var(X_3)}{\sqrt{cov(U,U)cov(V,V)}}$$

$$cov(U,U)=a_1^2var(X_1)+a_2^2var(X_2)+a_3^2var(X_3)$$
$$cov(V,V)=b_1^2var(X_1)+b_2^2var(X_2)+b_3^2var(X_3)$$

Therefore,

$$Corr(U,V)=\frac{a_1 b_1var(X_1)+a_2 b_2var(X_2)+a_3b_3var(X_3)}{\sqrt{\left(a_1^2var(X_1)+a_2^2var(X_2)+a_3^2var(X_3)\right)\left(b_1^2var(X_1)+b_2^2var(X_2)+b_3^2var(X_3)\right)}}$$

Now, if you want the correlation between $X_1+2X_2$ and $3X_1+aX_2$ to be zero, in other words, their covariance should be nil:

$$0=cov(X_1+2X_2,3X_1+aX_2)=3cov(X_1,X_1)+2acov(X_2,X_2)=3var(X_1)+2avar(X_2)$$

because $X_1$ and $X_2$ are independent, we have $cov(X_1,X_2)=0$

You choose 
$$a=-\frac{3var(X_1)}{2 var(X_2)}$$
We have $X_1,X_2$ and $X_3$ independent.

Let $U=a_1X_1+a_2X_2+a_3X_3$ and $V=b_1X_1+b_2X_2+b_3X_3$

$$Corr(U,V)=\frac{cov(U,V)}{\sqrt{var(U)var(V)}}=\frac{\sum_{k=1}^{3}{\sum_{j=1}^{3}{a_kb_jcov(X_k,X_j)}}}{\sqrt{cov(U,U)cov(V,V)}}$$

if $ k \ne j$, $cov(X_k,X_j)=0$

$$Corr(U,V)=\frac{a_1 b_1var(X_1)+a_2 b_2var(X_2)+a_3b_3var(X_3)}{\sqrt{cov(U,U)cov(V,V)}}$$

$$cov(U,U)=a_1^2var(X_1)+a_2^2var(X_2)+a_3^2var(X_3)$$
$$cov(V,V)=b_1^2var(X_1)+b_2^2var(X_2)+b_3^2var(X_3)$$

Therefore,

$$Corr(U,V)=\frac{a_1 b_1var(X_1)+a_2 b_2var(X_2)+a_3b_3var(X_3)}{\sqrt{\left(a_1^2var(X_1)+a_2^2var(X_2)+a_3^2var(X_3)\right)\left(b_1^2var(X_1)+b_2^2var(X_2)+b_3^2var(X_3)\right)}}$$

Now, if you want the correlation between $X_1+2X_2$ and $3X_1+aX_2$ to be zero, in other words, their covariance should be nil:

$$0=cov(X_1+2X_2,3X_1+aX_2)=3cov(X_1,X_1)+2acov(X_2,X_2)=3var(X_1)+2avar(X_2)$$

because $X_1$ and $X_2$ are independent, we have $cov(X_1,X_2)=0$

You choose 
$$a=-\frac{3var(X_1)}{2 var(X_2)}$$