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!

  • $\begingroup$ "the facts that sum of two uniform variable is again a uniform random variable"... that is not a fact at all. $\endgroup$ – drhab Dec 11 '17 at 14:20
  • $\begingroup$ Perhaps it is not correct? , let me edit to sum of two normal random variables is a normal random variable.Like I was trying to express that we use these type of logic while solving many problems so any similar kind of facts would be useful. $\endgroup$ – BAYMAX Dec 11 '17 at 14:26

$U:=\frac1{\sqrt2}(X_1-X_2)$ and $V:=\frac1{\sqrt2}(X_3-X_4)$ are independent and have standard normal distribution.

Then $W=U^2+V^2$ has chi-squared distribution with $2$ degrees of freedom.

  • $\begingroup$ Can we say something about the distribution of $U^2$ ? $\endgroup$ – BAYMAX Dec 11 '17 at 15:38
  • $\begingroup$ $U^2$ has chi-squared distribution with one degree of freedom ($k=1$). $\endgroup$ – drhab Dec 11 '17 at 15:44
  • $\begingroup$ So squaring a random variable following normal distribution gives us a random variable following Chi-squared distribution! , also there is a Probability and Statistics chatroom, I admit its not so packed though! $\endgroup$ – BAYMAX Dec 11 '17 at 16:32
  • $\begingroup$ It is for me time to go now so I will not enter the room. Maybe later. $\endgroup$ – drhab Dec 11 '17 at 16:37
  • $\begingroup$ Sure! at your convenience! $\endgroup$ – BAYMAX Dec 11 '17 at 16:58

Chi-squared on two degrees of freedom. $(X_1-X_2)/\sqrt 2$ and $(X_3-X_4)/\sqrt 2$ are iid $N(0,1)$ distributed; the sum of their squares is chi-squared distributed on two degrees of freedom.


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