# What is the distribution of a sum of a subset of probabilities, with each probability having the same distribution?

Suppose I have $k$ outcomes with probabilities, $p_i$, with $p_1+p_2+\dots+p_k=1$. Each probability has the same distribution. What would the distribution of a sum of probabilities be? For example, what would the distribution of $p_1+p_2$ be? The Irwin-Hall distribution is the distribution for the sum of uniformly distributed random variables. Each $p_i$ will be distributed between 0 and 1, but they won't be uniformly distributed. Does the Irwin-Hall distribution have anything to do with it?

I don't require a fully derived answer, but a kick in the right direction would help.

• This question appears to be off-topic because it is about statistics, and not academia – TCSGrad Apr 1 '14 at 3:53
• Moment generating functions might be useful although it depends on what the distribution is – Seamus O'Bairead Apr 1 '14 at 7:51
• I'm confused. You appear to be asking about the distribution of a sum of probabilities. However, probabilities are not random variables, so the question is not meaningful. If you are considering some random variables taking values between 0 and 1 which happen to sum to 1, then don't use the word probabilities, it is confusing. I think you can handle it as you would the sum of any sum of random variables, namely using convolution. Since there is a constraint on the sum, you could remove one random variable from there, so you would for example be working with $X_1$, $X_2$, $1 - X_1 - X_2$. – Faheem Mitha Apr 1 '14 at 12:32
• Of course, you would have to know the joint distribution of the $X_i$'s to make this work. – Faheem Mitha Apr 1 '14 at 12:35

The Irwin-Hall distribution is the distribution of a random variable defined as the sum of a number of independent random variables, each having a uniform distribution. It won't be useful here since the probabilities $p_i$ are not uniformly distributed nor independent (they need to sum to one).

The distribution that will suit your needs is the Dirichlet distribution. It is the multivariate generalization of the beta distribution, and has support $x_1, \ldots, x_n$ with $x_i \in [0, 1]$ for $i=1, \ldots, n$ and $\sum_{i = 1}^n x_i = 1$.

Let $\boldsymbol{X} = (X_1, \ldots, X_n) \sim \mbox{Dir}(\alpha_1, \ldots, \alpha_n)$, and suppose that one seeks the distribution of $X_1 + X_2$. By the aggregation property $$\boldsymbol{Y} = (X_1 + X_2, X_3, \ldots, X_n) \sim \mbox{Dir}(\alpha_1 + \alpha_2, \alpha_3, \ldots, \alpha_n) .$$ Since marginal distributions of Dirichlet distributions are beta distributions, we have $$X_1 + X_2 \sim \mbox{Beta}\left(\alpha_1 + \alpha_2, \sum_{i = 3}^n \alpha_i\right) .$$

Probabilities can be treated as random variables and have distributions of their own -if their exact value is unknown. For example, the Beta Distribution is a distribution suited to model the distribution of a probability, since in its standard formulation it ranges in the [0,1] interval.

Assume therefore that each $p_i$ follows a (common as per question) $Beta (a,b)$ (the Uniform $U(0,1)$ is a special case of the beta distribution). The $p_i$'s must satisfy the constraint

$$\sum_{i=1}^kp_i =1 \Rightarrow E\left[\sum_{i=1}^kp_i\right] =1 \Rightarrow k\frac {a}{a+b}=1$$

which determines the one parameter of the beta distribution as a function of the other and of the magnitude of $k$.

What you need next is, as a comment points out, to determine the joint distribution of the $p_i$'s. Are the $p_i$'s independent random variables?

Think (and I suggest you work further) that

$$\sum_{i=1}^kp_i =1 \Rightarrow \left(\sum_{i=1}^kp_i\right)^2 =1 \Rightarrow \sum_{i=1}^kp_i^2 + 2\sum_{i\neq j}p_ip_j = 1$$

$$\Rightarrow \sum_{i=1}^kE[p_i^2] + 2\sum_{i\neq j}E[p_ip_j] = 1 = k\frac {a}{a+b}$$