Suppose I a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random variable $S_n$ which follows a Poisson-Binomial distribution of parameter $(p_i)_{i=1}^n$ supported in $\{0,\dots,n\}$.

I would like to generalize this situation to a ``unit mass" population of independent Bernoullis. To describe this more formally, let $F$ be the cdf of an (absolutely continuous) probability distribution, supported in $[0,1]$. Take $F$ to describe the frequency of the possible values of the Bernoulli parameters within the population.

Define $S$ to be the fraction of successes within the population. Clearly,should take values in $[0,1]$. Is $S$ a well defined random variable? If yes, what is its distribution? If no, is there something close to what I am after?

Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random variable $S_n$ which follows a Poisson-binomial distribution of parameter $(p_i)_{i=1}^n$ supported in $\{0,\dots,n\}$.

I would like to generalize this situation to a ``unit mass" population of independent Bernoullis. To describe this more formally, let $F$ be the cdf of an (absolutely continuous) probability distribution, supported in $[0,1]$. Take $F$ to describe the frequency of the possible values of the Bernoulli parameters within the population.

Define $S$ to be the fraction of successes within the population. Clearly,should take values in $[0,1]$. Is $S$ a well defined random variable? If yes, what is its distribution? If no, is there something close to what I am after?

Suppose I a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random variable $S_n$ which follows a Poisson-Binomial distribution of parameter $(p_i)_{i=1}^n$ supported in $\{0,\dots,n\}$.

I would like to generalize this situation to a ``unit mass" population of independent Bernoullis. To describe this more formally, let $F$ be the cdf of an (absolutely continuous) probability distribution, supported in $[0,1]$. Take $F$ to describe the frequency of the possible values of the Bernoulli parameters within the population of Bernoullis.

Define $S$ to be the fraction of successes within the population. Clearly,should take values in $[0,1]$. Is $S$ a well defined random variable? If yes, what is its distribution? If no, is there something close to what I am after?

Suppose I a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random variable $S_n$ which follows a Poisson-Binomial distribution of parameter $(p_i)_{i=1}^n$ supported in $\{0,\dots,n\}$.

I would like to generalize this situation to a ``unit mass" population of independent Bernoullis. To describe this more formally, let $F$ be the cdf of an (absolutely continuous) probability distribution, supported in $[0,1]$. Take $F$ to describe the frequency of the possible values of the Bernoulli parameters within the population.

Define $S$ to be the fraction of successes within the population. Clearly,should take values in $[0,1]$. Is $S$ a well defined random variable? If yes, what is its distribution? If no, is there something close to what I am after?