Six random variables

Consider the closed interval [0,1]. In general, we can solve this by nested integrations, where you integrate over all the possibilities for the first value being in this range. Then integrating over all the values where the second value is in this range (less the first value), then integrating over the third value being in this range (less the first two values) etc. until we have integrated over all the variables. The value of that integral will represent the final probability.

For example for six variables, $x_i$ and $u, v, w, x, y$ and $z$ defined as this:

• $u = 1 - x_1$
• $v = 1 - x_1 - x_2$
• $w = 1 - x_1 - x_2- x_3$
• $x = 1 - x_1 - x_2- x_3- x_4$
• $y = 1 - x_1 - x_2- x_3- x_4 - x_5$
• $z = 1 - x_1 - x_2- x_3- x_4 - x_5 - x_6$

we would have something like:

$P = \int_0^1 \int_0^u \int_0^v \int_0^w \int_0^x \int_0^y \, dz \, dy \, dx \, dw \, dv \, du$

$P = \int_0^1 \int_0^u \int_0^v \int_0^w \int_0^x y \, dy \, dx \, dw \, dv \, du$

$P = \int_0^1 \int_0^u \int_0^v \int_0^w \dfrac{x^2}{2} \, dx \, dw \, dv \, du$

$P = \int_0^1 \int_0^u \int_0^v \dfrac{w^3}{6} \, dw \, dv \, du$

$P = \int_0^1 \int_0^u \dfrac{v^4}{24} \, dv \, du$

$P = \int_0^1 \dfrac{u^5}{120}\, du$

$P = \dfrac{1}{720}$

$P = \dfrac{1}{6!}$

$1+720=\boxed{721}$

The sum of n independent random variables with uniform distribution is the Irwin-Hall distribution, whose cumulative distribution function is: Let X be the sum of the 6 random variables, then: P(X<1)= $\frac{1}{6!}*((-1)^0*(6C0)*(1-0)^6+(-1)^1*(6C1)*(1-1)^6)$

P(X<1)= $\frac{1}{720}$

a+b=720+1= 721

Not a creative solution thought :(

let us consider no line in 0 to 1 is made up of n points (mark them 1 to n) choosing any 6 pts we have way n^6 choice now let us find no of ways x1+x2+x3+x4+x5+x6<n(marked values) we get it as (n-1)C6 so divide and find limit n tends to infinity. :P

Solving for all $n$ (here $n=6$ ) with $S_n = \sum_{i=1}^n x_i$ :

$\int_{\forall i \in [0,n] \, 0 < x_i < 1, \,S_n < 1} dx_1 ... dx_n \\ = \int_{\forall i \in [0,n-1] \,0 < x_i < 1, \, 0 < x_n < 1 - S_{n-1} , \, S_{n-1} < 1 } dx_1 ... dx_n \\ = \int_{\forall i \in [0,n-1] \, 0 < x_i < 1 , \, S_{n-1} < 1 }(1 - S_{n-1})^1 dx_1 ... dx_{n-1} \\ = \int_{\forall i \in [0,n-2] \, 0 < x_i < 1, \, S_{n-2} < 1 }\frac{(1 - S_{n-1})^2}{2} dx_1 ... dx_{n-2} \\ =... = \frac{1}{n!}$

So the answer is $n! +1$ , here $721$ .