Sum of random number of variables

Probability Level 4

Let $x_1,x_2,x_3, \ldots$ be sequence of numbers randomly picked from the interval $[0,1]$ with uniform distribution. define $S=x_1+\cdots+x_n\mid x_1+\cdots +x_{n}\geq 1 \land x_1+\cdots+x_{n-1}<1.$

The expected value $E\left[S\right]$ can be written as $\dfrac{ae}{b}$ , where $a$ and $b$ are coprime integers. Find $a+b$

Inspiration 1 and Inspiration 2 .

The answer is 3.

1 solution

Aareyan Manzoor
Jun 25, 2017

this solution is wrong, see comments

the trick is to use iterated expectations. first we find the conditional expectation: $E\left[S|N=n\right]=E\left[x_1+...+x_n\right]=E\left[x_1\right]+...+E\left[x_n\right]=nE\left[x_1\right]=\dfrac{n}{2}\\ \to E\left[S|N\right]=\dfrac{N}{2}$ we know by iterated expectations that $E\left[S\right]=E\left[E\left[S|N\right]\right]=\dfrac{1}{2} E\left[N\right]=\dfrac{e}{2}$ note the result for $E\left[N\right]$ comes from this problem

Hm, I do not think that $E[ S | N = n ] = \frac{n}{2}$ . You are skipping the information that $x_1 + \ldots + x_{n-1} \geq 1$ .

As an explicit counter example, $E [ S |n = 1 ]$ would be 0, since this happens with probability 0.

Also, it should be clear to you that $E [ S | n = 2 ] > 1$ , since $S > 1$ almost certainly.

Calvin Lin Staff - 3 years, 11 months ago

that does seem to be the case. although a monte-carlo simulation does reveal an answer close to $e/2$ . i tried making an integral for expected value when N=n but it does become tedious. i guess i will wait for a proper solution to this.

Aareyan Manzoor - 3 years, 11 months ago

Sum of random number of variables

The answer is 3.

1 solution

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