Choose a random number between $0$ and $1$ and record its value. Keep doing it until the sum of the numbers exceeds $1$. How many tries do we need?

Solution 1:

Here is a (perhaps) more elementary method. Let $X$ be the amount of numbers you need to add until the sum exceeds $1$. Then (by linearity of expectation):

$$ \mathbb{E}[X] = 1 + \sum_{k \geq 1} \Pr[X > k] $$

Now $X > k$ if the sum of the first $k$ numbers $x_1,\ldots,x_k$ is smaller than $1$. This is exactly equal to the volume of the $k$-dimensional set:

$$ \left\{(x_1,\ldots,x_k) : \sum_{i=1}^k x_i \leq 1, \, x_1,\ldots,x_k \geq 0\right\}$$

This is known as the $k$-dimensional simplex. When $k = 1$, we get a line segment of length $1$. When $k = 2$, we get a right isosceles triangle with sides of length $1$, so the area is $1/2$. When $k=3$, we get a triangular pyramid (tetrahedron) with unit sides, so the volume is $1/6$. In general, the volume is $1/k!$, and so

$$ \mathbb{E}[X] = 1 + \sum_{k \geq 1} \frac{1}{k!} = e. $$

Solution 2:

Assuming the numbers come from a uniform distribution over $[0,1]$ and that the trials are independent, here is an outline (this is example 7.4 4h. in Sheldon Ross' A First Course in Probability, sixth edition):

1. Let $X_i$ be the number obtained on the $i$'th trial.

2. For $0\le x\le1$, let $Y(x)$ be the minimum number of trials needed so that the sum of the $X_i$ exceeds $x$. Set $e(x)=\Bbb E [Y(x)]$.

3. Compute $e(x)$ by conditioning on the value of $X_1$: $$\tag{1} e(x)=\int_0^1 \Bbb E [ Y(x) | X_1=y]\, dy. $$

Here, use the fact that $$\tag{2}\Bbb E [ Y(x) | X_1=y] = \cases{1,& $y>x$\cr 1+e(x-y),& $y\le x $}.$$

Substitution of $(2)$ into $(1)$ will give $$\tag{3} e(x)=1+\int_0^x e(u)\,du. $$

4. Solve equation $(3)$ (by differentiating both sides with respect to $x$ first) for $e(x)$.

5. You wish to find $e(1)$.

Solution 3:

Here is another method. This is exactly the same technique Yuval has given above but with an added step that goes through computing the pmf first before computing the expectation. Hopefully, this will help you understand the problem from a slightly different angle.

Let us denote by $N$ the number of random variables we need to add for the sum to exceed $1$. We first find the distribution (probability mass function) of $N$. The easiest way to do this is by computing $P(N > n)$ for $n=1,2,3,\dots$. Once we know this, we can compute $P(N = n) = P(N > n-1) - P(N > n)$.

The event that $(N > n)$ in plain English says that the sum of the first $n$ uniform random variables did not exceed $1$. The probability of this event is therefore $P(N > n) = P(U_1 + U_2 + \dots + U_n < 1)$. This can be calculated by a standard multi-dimensional integral to be $\frac{1}{n!}$. If you don't want to use calculus, one can justify this result geometrically but integration is perhaps the most natural approach to evaluate this probability. Therefore, we have $P(N = n) = \frac{1}{(n-1)!} - \frac{1}{n!} = \frac{n-1}{n!}$ for $n=1,2,3\dots$.

Once we know the pmf of N, we calculate

$E(N) = \sum_{n=1}^{\infty} n P(N=n) = \sum_{n=1}^{\infty} \frac{n(n-1)}{n!} = \sum_{n=0}^{\infty} \frac{1}{n!} = e$.

Solution 4:

The solutions presented here are short and elegant, but I do not find them very intuitive. Here is a longer solution that I find more intuitive.

Restating the problem

Let us sample values, $x_i$, from a uniform distribution between 0 and 1 until their sum is greater than 1,

$$ x_1 + x_2 + x_3 + \ldots + x_n > 1\;, $$

where the $n$-th value is the first value such that the sequence sums to a number greater than 1. This means that the sum of the sequence up to the $(n-1)$-th term has to be lower or equal to 1,

$$ x_1 + x_2 + x_3 + \ldots + x_{n-1} \leq 1\;. $$

What is the expected value of the sequence length, $n$?

Solution

The expected value of the sequence length, $n$, is, by definition,

$$ \mathbb{E}(n) = \sum_{n=1}^{\infty} n P(n)\;, $$

where $P(n)$ is the probability that the sequence terminates with the $n$-th element. So, by finding $P(n)$, we should be able to calculate $\mathbb{E}(n)$.

To calculate the probability, $P(n)$, let us consider the following facts:

• The sequence of elements up to the $(n-1)$-th elements has to sum up to a value, $x \leq 1$, that is, $\sum_{i=1}^{n-1} x_i = x \leq 1$.
• Adding the $n$-th element to $x$ has to lead to a value higher than 1, that is, $x + x_n > 1$.

As a first step, let us consider a sequence whose first $(n-1)$ values sum to $x$. This happens with probability,

$$ P(x_1 + \ldots + x_{n-1} = x) = \int_0^1 \mathrm{d}x_1 \ldots \int_0^1 \mathrm{d}x_{n-1} \, \delta(x_1 + \ldots + x_{n-1} = x) \; , $$

where $\delta(x)$ is the Dirac delta. Given that the sum of the first $(n-1)$ elements is $x \leq 1$, the probability that summing the $n$-th element results in a value greater than 1 is,

$$ P(x + x_n > 1) = P(x_n > 1 - x) = x $$

Hence, the probability that the first $(n-1)$ elements sum to $x < 1$ and that the $n$-th elements leads to a sum higher than 1 is the product of the two probabilities just derived,

$$ P(x_1 + \ldots + x_{n-1} = x) \, P(x + x_n > 1) \;. $$

Here, $x$ is fixed. To consider all cases, we have to sum over all the values of $x \leq 1$,

$$ P(n) = \int_0^1 \mathrm{d}x \, P(x_1 + \ldots + x_{n-1} = x) \, P(x + x_n > 1) \;. $$

Substituting the expressions of the probabilities, we have,

$$ P(n) = \int_0^1 \mathrm{d}x \, \int_0^1 \mathrm{d}x_1 \ldots \int_0^1 \mathrm{d}x_{n-1} \, \delta(x_1 + \ldots + x_{n-1} = x) \, x \;. $$

We integrate over $x$ to find,

$$ P(n) = \int_0^1 \mathrm{d}x_1 \ldots \int_0^1 \mathrm{d}x_{n-1} \, \theta(1 - x_1 - \ldots - x_{n-1}) \, (x_1 + \ldots + x_{n-1}) \;, $$

where $\theta$ is the Heaviside step function. The equation is symmetric in all the $x_i$'s, meaning that it simplifies to,

$$ P(n) = (n - 1) \int_0^1 \mathrm{d}x_1 \ldots \int_0^1 \mathrm{d}x_{n-1} \, \theta(1 - x_1 - \ldots - x_{n-1}) \, x_{n-1} \;. $$

We can rearrange the equation into,

$$ P(n) = (n - 1) \int_0^1 \mathrm{d}y \, y \left[ \int_0^1 \mathrm{d}x_1 \ldots \int_0^1 \mathrm{d}x_{n-2} \, \theta((1 - y) - x_1 - \ldots - x_{n-2}) \right]\;. $$

The multi-dimensional integral in the square brackets is the volume of the $(n-2)$-dimensional simplex with edge of length $(1-y)$, which is $(1-y)^{n-2} / (n - 2)!$. To understand this, note that the volume of a $k$-dimensional simplex with an edge of unit length is $1/k!$. Shortening the edge in each dimension by a factor of $z$ leads to a reduction of the volume by a factor $z^k$. Hence the volume of a $k$-dimensional simplex with edge of length $z$ is $z^k/k!$. Substituting $k=n-2$ and $z=1-y$, we get the formula stated above. We substitute this formula into the equation of $P(n)$ to obtain,

$$ P(n) = (n - 1) \int_0^1 \mathrm{d}y \, y \left[ \frac{(1-y)^{n-2}}{(n - 2)!} \right]\;. $$

We move the denominator outside of the integral and substitute $y \rightarrow 1 - y$,

$$ P(n) = \frac{n - 1}{(n - 2)!} \int_0^1 \mathrm{d}y \, (1 - y) y^{n-2} = \frac{n - 1}{(n - 2)!} \frac{1}{n(n-1)} = \frac{n - 1}{n!}\;. $$

Finally, we substitute this expression for the probability, $P(n)$, into the equation of the expectation value of $n$ and obtain,

$$ \mathbb{E}(n) = \sum_{n=1}^{\infty} n P(n) = \sum_{n=1}^{\infty} n \frac{n - 1}{n!} = \sum_{n=2}^{\infty} n \frac{n - 1}{n!} = \sum_{n=2}^{\infty} \frac{1}{(n-2)!} = \sum_{n=0}^{\infty} \frac{1}{n!} = e\;, $$

Hence, the expected value of the sequence length, $n$, is $e$.