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If $X,Y,Z$ are independent standard normal random variables, compute P(3X+2Y<6Z-7).

One way is to evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{(6z-2y-7)/3}\frac{\exp(-(x^2+y^2+z^2)/2)}{2\pi\sqrt{2\pi}}dxdydz.$ But I don't know how to calculate this.

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    See my answer below for how to combine the variances.2012-04-21

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If you have two independent normally distributed random variables, then the distribution of their sum is also normally distributed, where:

$\mu=\mu_1+\mu_2$

$\sigma^2=\sigma_1^2+\sigma_2^2$

You can use this to solve your problem without integrals, other than one evaluation of the error function. Actually, you don't even need that, since the $7$ turns out to give you a special result for which you probably already know the result of applying the error function.

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    I have updated the answer to say "independent". I realize that that is slightly stronger than "jointly", but is likely more widely understood than "jointly". You are right that I had to say something more about the relation between the two distributions to make the statement correct.2012-04-21