## Week 1: Estimators
## Code from class

## Monte Carlo Example

# define a function for the sample variance
sv <- function(y) { length(y) / (length(y)-1) * mean(y)^2 }
# create a loop to compute and store 1,000 simuated sample variances
# this will be from 50 random y values
result <- numeric(500)
for (i in 1:500) { result[i] <- sv(rnorm(50)) }
# plot the result, after sorting the computed sample variances
plot(sort(result), type="l", ylab="Computed sample variance")
# now check whether sampling distribution matches a Chi^2
chisq.test(result)


## Birthday problem analysis

lbdp <- function(n) {
  1 - exp(lfactorial(365) - lfactorial(365-n) - n*log(365))
}

x <- 1:60
plot(x,lbdp(x))

plot(x,lbdp(x),
     xlab="Number of people",ylab="Probability of same birthday")
abline(h=.5, lty="dashed", col="red")