# “Weighted” Poisson binomial distribution

I have stones of different weights. For each stone, I flip the same fair coin. If it's heads, I add the stone's weight to a running total. Given the weights, can I find the distribution for the total score after I flip all coins?

For example: if I have $n = 3$ stones of weights $4, 5.5$, and $10$, and the coin flips are HHT, then the sum is $9.5$.

My question seems the same as this one, and the distribution seems like a modified Poisson binomial distribution.

• Are the weights all known fixed constants, or are they drawn from some distribution? In either case it's possible (but not necessarily convenient for very large n). You have other options (simulation, approximation) if you want to answer particular questions about the distribution. – Glen_b -Reinstate Monica Sep 17 '18 at 4:07

The difference from the linked post is that there the "weights" are integers, while here they are general positive constants. This is not similar to a poisson distribution, more with a binomial. In the general case without some restrictions on the individual weights $$x_i$$, this will be a discrete distribution with equal probabilities on each of $$2^n$$ values. For small $$n$$ we can do a complete tabulation, in general only approximations will be practical.
Some notation. Let the known stone weight be positive numbers $$x_1, \dotsc, x_n$$. The coin toss result is random variables $$W_1, W_2\dotsc, W_n$$ independent with probability $$p=0.5$$ of being 1, else 0. The total random weight is $$W=\sum_1^n W_i x_i$$. This is the same situation as the usual model of sampling finite populations, so look at the tags or .
Some approximations. Calculating expectation and variance gives $$\DeclareMathOperator{\E}{\mathbb{E}}\DeclareMathOperator{\Var}{\mathbb{V}ar} \E W =\frac12 \sum x_i \\ \Var W = p(1-p) \sum x_i^2.$$ From this a normal approximation can be calculated.
A much better approximation is the saddlepoint approximation, which needs the moment generating function (mgf), which can be calculated as $$M(t) = \E e^{t W} = \E e^{t\sum_i x_i W_i}=\prod_1^n \left\{ p e^{tx_i}+(1-p)\right\}$$ which is indeed similar in form to the binomial mgf. An example will be very similar to the bootstrap example given in How does saddlepoint approximation work?.