I have a variable which is a linear combination of two other variables, each one following an N(0,1) distribution.

I need to compute the threshold of the distribution of this combination variable (to compute the Z for a specific probability).

Does anyone know which function should I use, i.e. which package is available to perform this task?

  • $\begingroup$ I think you should update your title to reflect the fact that you're looking for a quantile (given a probability value) rather than a probability value. (If I understand the body of your question correctly.) $\endgroup$
    – chl
    Dec 15, 2011 at 11:29

1 Answer 1


If you know what the coefficients in the linear combination, and the two random variables are independent, this is simple. We have:

$$X=a_1Z_1+a_2Z_2\sim N\left(0,\sqrt{a_1^2+a_2^2}\right)$$

This is an elementary statistics result, and can be shown by the method of characteristic functions, or convolutions (and other methods...). So you can use a normal tables, and any statistical package has normal tables - take your pick. For example, in excel you can use norm.inv(...), in R you can use qnorm(..), in SAS you can use QUANTILE("NORMAL",..) etc. etc.

  • 2
    $\begingroup$ In fact, the same method works if $Z_1$ and $Z_2$ are jointly normal with correlation coefficient $\rho$ (instead of being independent) with the minor twist that now $$X=a_1Z_1+a_2Z_2\sim N\left(0,\sqrt{a_1^2+a_2^2 +2\rho a_1a_2}\right).$$ Of course, $\rho = 0$ if $Z_1$ and $Z_2$ are independent and so the bells-and-whistles formula given above simplifies to the one given in probabilityislogic's answer in this special case. $\endgroup$ Dec 15, 2011 at 13:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.