Overdispersion in learning to play darts

Suppose a player plays T=100 games of darts, each game consisting of ten throws. Assume the player gets better at hitting the dartboard over time, learning in an S-curve over the 100 games, but holding success probability constant over the course of a game.



In [183]:

    
n=10
T = 100
x = rep(0,T);
linear = (-50:50)/10
ps=1/(1+exp(-linear))
for(i in 0:T)
{
  x[i]<-rbinom(1,n,ps[i]);
}
par(mfrow=c(1, 2)); plot(x, main="no. of hits", xlab="Game number"); plot(ps,main="prob of hit", xlab="Game number")









    



Warning message:
In rbinom(1, n, ps[i]): NAs produced



In [184]:

    
phat = mean(x)/n; phat









    Out[184]:




0.48

Assuming that the number of hits in each of the 100 games is binomial(n,.5), we'd expect a variance of



In [185]:

    
n*phat*(1-phat)









    Out[185]:




2.496

But the observed actual variance is higher:



In [186]:

    
var(x)









    Out[186]:




15.7373737373737



In [187]:

    
hist(x)



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