# 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.

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In :

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")

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

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In :

phat = mean(x)/n; phat

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Out:

0.48

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

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In :

n*phat*(1-phat)

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Out:

2.496

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But the observed actual variance is higher:

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In :

var(x)

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Out:

15.7373737373737

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In :

hist(x)

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In [ ]:

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