Distribution for number of returned items per day I am interested in estimating the percentage of returned/faulty items out of all items purchased in a shop per day (assumed that the items are returned the same day as purchased and a different number of items are purchased per day). I have a dataset that shows me the purchased and returned items every day in a month. What would be an appropriate distribution to model this scenario? Should I use Poisson and assume rates instead of counts? Or would a Binomial be more appropriate?
 A: Poisson rate $\lambda$ for number $B$ of purchases each day. Conditionally, on $B = b$ purchases in a day,
the number of returns is $R \sim \mathsf{Binom}(b, p),$ where $p$ is the probability
an item will be returned.
Suppose we have data for $250$ days, $\lambda = 5,\; p = 0.1.$ Here is a simulation
of purchases and returns:
set.seed(2020)
b = rpois(250, 5)
r = rbinom(250, b, .1)
summary(r)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   0.00    0.00    0.00    0.52    1.00    3.00 

MAT = rbind(b,r);  MAT[,1:10]  # first ten days
  [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
b    6    4    6    5    3    2    3    4    0     6
r    1    3    1    1    1    0    1    0    0     0

hist(r, prob=T, br=(-1:5)+.5, col="skyblue2", main="Simulated Dist'n of Returns")
  points(0:5, dpois(0:5, .5), col="red")


I will leave it to you to show that $R \sim\mathsf{Pois}(\lambda_r = p\lambda),$
as roughly illustrated in the plot above of simulated returns for 250 days.
You can estimate $\lambda$ by the average number of purchases per day, and $p$ as the overall fraction of returned items.
