If $X_{i} , i = 1,2,3,4$ are independent standard normal random variable.
Then the distribution of $W = \frac{1}{2} ((X_{1}-X_{2})^2 + (X_{3} - X_{4})^2)$ ?
Can we generalize these type of questions?
Like I know that sum of two normal random variables is again a normal random variable but what can we say about the product?
Also are there any other common facts one should know about the arithmetic of random variables following some other distributions? Like how can one intuitively think about the facts that sum of two Normal random variables is again a normal random variable? or like a distribution whose mean and variance are equal? - its Poisson distribution.
Any other list of such facts based on experience will be very much helpful!