# How to compute a probability threshold for a linear combination of two variables ~ N(0,1)?

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?

• 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.) – chl Dec 15 '11 at 11:29

$$X=a_1Z_1+a_2Z_2\sim N\left(0,\sqrt{a_1^2+a_2^2}\right)$$
• 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. – Dilip Sarwate Dec 15 '11 at 13:48