### MEETING 2

### LING280 CLASS TRANSCRIPT
### January 10, 2013

### CONTINUING FROM MEETING 1
### CUMULATIVE DENSITY FUNCTION

# What is the following command going to do?
# Hint: give the command `?plot` to find the arguments to plot()
# load the plurals data ...
# read.csv("/path/to/plurals.csv")->plurals
# my.sample <- sample(plurals$RT, 50)

plot(quantile(my.sample, seq(0, 1, 0.05)), seq(0, 1, 0.05))
abline(h = 0.5, v = median(my.sample))

# Plot another quantile (abline h = ...)
> ...

# Histogram
hist(my.sample)
hist(my.sample, breaks=5)
hist(my.sample, breaks=15)
hist(my.sample, breaks=seq(0, 1500, 50), col="blue", density=25) -> my.hist

# inspect the hist object
> ...

# 
cumsum(my.hist$counts) -> my.hist.cumsum

# 
barplot(my.hist.cumsum, 
	ylim=c(0,50), density=35, col="blue", angle=-45, names=my.hist$mids)
barplot(my.hist$counts, 
		ylim=c(0,50), density=45, col="red", angle=45, add=1)
		
# Empirical CDF/PDF
> ...
> my.cumulative.fraction <- ...
> ...


###
### BINOMIAL THEOREM, BINOMIAL DISTRIBUTION
###

rbinom(1 , 10, 0.5)
rbinom(10, 1, 0.5) 

# WHAT ...
#
# Args:  n
#	 size
#	 prob
# Returns:

# Sample 'p'
sum( rbinom( 10, 1, 0.5 ) ) / 10
sum( rbinom( 100, 1, 0.5 ) ) / 100
sum( rbinom( 10000000, 1, 0.5 ) ) / 10000000

# Simulate a large number of experiments
my.experiments <- rbinom(100000, 40, 0.5)
hist(my.experiments, breaks=0:40, freq=FALSE)


### COMPOUND EVENTS

### 4 CHOOSE 1
choose(4, 1)
### 4 CHOOSE 2
choose(4, 2)

#
choose(4, 0:4)

#
for(i in 1:10){
  print(choose(i, 0:i))
}


### OK, now what about for 40 events?
num.outcomes <- choose(40, 0:40)

### How many total possibilities are there?
> ....

### PDF
hist(my.experiments,breaks=0:40,freq=FALSE)
lines(0:40, ... , pty = 'l', col="red")

### CDF
> ...


### Deriving the binomial distribution

p <- ...

### COMPUTE THE DISTRIBUTION
binom.distr.40 <- choose(40, 0:40) *  p

### SAME PLOT AS ABOVE
hist(my.experiment, breaks=0:40, freq=FALSE)
lines(binom.distr.40)

### == 1
sum( ... )

### WHAT IS THE EXACT PROBABILITY OF 20 occurring?

> ....

### SUPPOSE THE LIKELIHOOD IS NOT p = 0.5, but p = 0.75

> .... 

### p = 0.75, 40 events, likelihood of count being 30

> ....

### p = 0.75, 4 events, likelihood of count being 3

> ....

### p = 0.75, 400 events, likelihood of count being 300

> ....

### INTERVALS/RANGES

n.events <- 40
p.event <- 0.7

sim.events<- rbinom(10000, 40, 0.7)

probs <- dbinom(0:n.events, n.events, p.event)

plot(0:40, probs, col="red", lwd=3, pty='l')
hist(sim.events, breaks=0:40, freq=FALSE, col="grey", density=20, add=1)

abline( v = mean(sim.events) )

###

###  +/- 1
sum(probs[27:29])
abline( v = c(27, 29), col="blue",lwd=2) 

###  +/- 2
sum(probs[26:30])
abline( v = c(26, 30), col="green", lwd=2) 

###  +/- 6
sum(probs[22:34])	
abline( v = c(25, 31), col="purple", lwd=2) 


#### TAKE HOME CHALLENGE
#### REPLICATE FIGURE 2.8 in V&B WITHOUT COPYING THEIR CODE VERBATIM