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.
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 survey or survey-sampling.
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?.
This is a particular case of Generalized Poisson Binomial distribution with the difference that each trial is identically distributed.
There is a recent R package PoissonBinomial implementing a number of new high-performance algorithms to compute quantities for this distribution.
