September 26, 2018

Announcements

• Our Wednesday, October 31st meetup has been moved to Tuesday, October 30th.

Meetup Presentations

• Suma Gopal (1.9)
• John Hancock (1.69)
• Calvin Wong (2.5)

Coin Tosses Revisited

coins <- sample(c(-1,1), 100, replace=TRUE)
plot(1:length(coins), cumsum(coins), type='l')
abline(h=0)

cumsum(coins)[length(coins)]
## [1] -12

Many Random Samples

samples <- rep(NA, 1000)
for(i in seq_along(samples)) {
coins <- sample(c(-1,1), 100, replace=TRUE)
samples[i] <- cumsum(coins)[length(coins)]
}
## [1]  -8   8  -2 -10  -8   6

hist(samples)

Properties of Distribution

(m.sam <- mean(samples))
## [1] 0.162
(s.sam <- sd(samples))
## [1] 9.883088

Properties of Distribution (cont.)

within1sd <- samples[samples >= m.sam - s.sam & samples <= m.sam + s.sam]
length(within1sd) / length(samples)
## [1] 0.677
within2sd <- samples[samples >= m.sam - 2 * s.sam & samples <= m.sam + 2* s.sam]
length(within2sd) / length(samples)
## [1] 0.951
within3sd <- samples[samples >= m.sam - 3 * s.sam & samples <= m.sam + 3 * s.sam]
length(within3sd) / length(samples)
## [1] 0.999

Standard Normal Distribution

$f\left( x|\mu ,\sigma \right) =\frac { 1 }{ \sigma \sqrt { 2\pi } } { e }^{ -\frac { { \left( x-\mu \right) }^{ 2 } }{ { 2\sigma }^{ 2 } } }$

x <- seq(-4,4,length=200); y <- dnorm(x,mean=0, sd=1)
plot(x, y, type = "l", lwd = 2, xlim = c(-3.5,3.5), ylab='', xlab='z-score', yaxt='n')

What's the likelihood of ending with 15?

pnorm(15, mean=mean(samples), sd=sd(samples))
## [1] 0.9333678

What's the likelihood of ending with 15?

1 - pnorm(15, mean=mean(samples), sd=sd(samples))
## [1] 0.06663219