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How can I find this probability $P(X<Y)$ ? knowing that X and Y are independent Normal random variables.

$X$~$N(m_1 ,v_1)$ (with mean m1 and variance v1)

$Y$~$N(m_2 ,v_2)$

I know that $$ \Pr \left[ X < Y \right] = \int F_X \left( y \right) f_Y \left( y \right) \mathrm{d} y $$ but I cannot solve it for normal random variable.

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up vote 3 down vote accepted

The key theorem is the following: Let $X$ and $Y$ be independent normal, with means $\mu$ and $\nu$, and variances $\sigma^2$ and $\tau^2$ respectively. Let $W=X-Y$. Then $W$ is normally distributed, with mean $\mu-\nu$, and variance $\sigma^2+\tau^2$.

Now it is easy to compute $\Pr(W\lt 0)$.

If $\mu=\nu$, we don't need the above machinery. For then the random variable $W$ has distribution which is symmetric about $0$, so the probability is $1/2$.

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