# Question 1

n = 15; mu1 = 5; sigma1 = 2
m = 20; mu2 = 5; sigma2 = 3

x.samp = rnorm(n, mu1, sigma1)
y.samp = rnorm(m, mu2, sigma2)

xy.samp = c(x.samp, y.samp)

grp = c(rep(1,n),rep(2,m))

xy = data.frame(xy = xy.samp)
xy$grp = as.factor(grp)

attach(xy)

boxplot(xy ~ grp, data = xy)

# one-sided independent two sample t-test with unequal variance

t.test(xy.samp[grp=="2"], xy.samp[grp=="1"],
       alternative = "two.sided", mu = 0, var.equal = FALSE)

# Question 2

n = 15; mu1 = 5; sigma1 = 2
m = 20; mu2 = 7; sigma2 = 3

x.samp = rnorm(n, mu1, sigma1)
y.samp = rnorm(m, mu2, sigma2)

xy.samp = c(x.samp, y.samp)

grp = c(rep(1,n),rep(2,m))

xy = data.frame(xy = xy.samp)
xy$grp = as.factor(grp)

attach(xy)

boxplot(xy ~ grp, data = xy)

# one-sided independent two sample t-test with unequal variance

t.test(xy.samp[grp=="2"], xy.samp[grp=="1"],
       alternative = "two.sided", mu = 0, var.equal = FALSE)

# Distribution of the p-value

B = 10000

p.val = numeric(B)

for(i in 1:B){
	x.samp = rnorm(n, mu1, sigma1)
	y.samp = rnorm(m, mu2, sigma2)

	p.val[i] = t.test(x.samp, y.samp,
       alternative = "two.sided", mu = 0, var.equal = FALSE)$p.value

}

hist(p.val)

# Poisson sampling

n = 15; m = 20

set.seed(1)

mux = 5
muy = 7

x = rpois(n,mux);x;length(x);mean(x);var(x)
y = rpois(n,muy);y;length(y);mean(y);var(y)

mean(y)/mean(x)

plot(y,x)