CS112 14.1 game

# NOTE TO INSTRUCTOR: READ ENTIRE FILE BEFORE CLASS
# RUN CODE FROM LINE 49 TO END BEFORE CLASS

# CONSIDER THE FOLLOWING GAME...
# IMAGINE YOU WANT TO ESTIMATE THE UNDERLYING PARAMETER VALUE
# OF A BERNOULLI DISTRIBUTION (i.e., THE PROBABILITY OF A "1"
# INSTEAD OF A "0"...)

# YOU CAN PULL THE "CORR" LEVER AND OBTAIN 50 CORRELATED DATA 
# POINTS FROM THE BERNOULLI DISTRIBUTION

# OR YOU CAN PULL THE "UNCORR" LEVER AND OBTAIN 50 UNCORRELATED DATA
# POINTS FROM THE BERNOULLI DISTRIBUTION.

# NOTICE THAT BOTH LEVERS PRODUCE RESULTS THAT HAVE THE SAME 
# EXPECTED VALUE AND THEREFORE THE SAME VARIANCE & STD DEVIATION
# BECAUSE IN A BERNOULLI DISTRIBUTION, THE MEAN DETERMINES THE VARIANCE

# WHICH IS A BETTER CHOICE (TO LEARN ABOUT THE TRUE PARAMETER VALUE)?
corr.lever()
uncorr.lever()

sd(corr.lever())
sd(uncorr.lever())

plot(density(corr.lever()))
plot(density(uncorr.lever()))

# There are only 6 different outcomes you can get.
yy <- c()
for(i in 1:10000) {yy[i] <- mean(corr.lever())}
hist(yy)

# run 1000 experiments (simulated trials)
for(i in 1:1000)
{
  storage.corr[i] <- mean(corr.lever())
}

for(i in 1:1000)
{
  storage.uncorr[i] <- mean(uncorr.lever())
}

# Was the standard deviation of the means derived from uncorrelated data
# LOWER than the standard deviation of the means derived from correlated data?
# Across all the simulated trials (for the whole shebang)?
sd(storage.uncorr) < sd(storage.corr)

# classic standard error formula: sqrt( p*(1-p) ) / sqrt(n)
cat("uncorrelated data standard deviation:", sd(storage.uncorr))

# classic standard error formula does not work for correlated data
cat("correlated data standard deviation:", sd(storage.corr)) # standard error

# CODE BELOW: 
# RUN BEFORE CLASS TO LOAD FUNCTIONS IN MEMORY. 
# SHOW STUDENTS AFTER GAME ENDS.
corr.lever <- function()
  {
  return(c(
    rep(rbinom(n = 1, size = 1, prob = 0.5), 10),
    rep(rbinom(n = 1, size = 1, prob = 0.5), 10),
    rep(rbinom(n = 1, size = 1, prob = 0.5), 10),
    rep(rbinom(n = 1, size = 1, prob = 0.5), 10),
    rep(rbinom(n = 1, size = 1, prob = 0.5), 10)))
}

# "PULL THE *UNCORR* LEVER"
# uncorrelated data
uncorr.lever <- function()
{
  return(rbinom(n = 50, size = 1, prob = 0.5))
}

# WHICH LEVER GIVES YOU BETTER INFORMATION?
# WHY?