$x_i \sim \mathcal N(\mu,\sigma)$ $(i = 1,2,...n)$ are i.i.d normal distributed, $a$ is a constant, $a>0$, how to calculate the probability:
$$P(x_1 < a, x_1 + x_2 < a, x_1 + x_2 + x_3 < a,..., x_1 + ... + x_{n-1} < a, x_1 + x_2 + ... + x_n >= a)$$
maybe we can translate to:
$z_i \sim \mathcal N(0,1)$ $(i = 1,2,...n)$ are i.i.d normal distributed, $a$ is a constant, we denote
$$a_n = \frac{a - n \mu}{\sigma\sqrt{n}}$$
the probability may transformed into:
$P(z_1 < a_1, z_1 + z_2 < a_2, z_1 + z_2 + z_3 < a_3,..., z_1 + ... + z_{n-1} < a_{n-1}, z_1 + z_2 + ... + z_n >= a_n)$
now we denote:
$P_1 = P(z_1 < a_1, z_1 + z_2 < a_2, z_1 + z_2 + z_3 < a_3,..., z_1 + ... + z_{n-1} < a_{n-1}, z_1 + z_2 + ... + z_n >= a_n)$
and
$P_n = P(z_1 < a_1, z_1 + z_2 < a_2, z_1 + z_2 + z_3 < a_3,..., z_1 + ... + z_{n-1} < a_{n-1}, z_1 + z_2 + ... + z_n < a_n)$
that means
$P_{n-1} = P_1 + P_n$
so the problem can be solved by get $P_n$, but how to calculate it?