1

vaild_number = 0
total_times = 0
while (total_times <= 100000) {
    if(sum(sample(x=c(0,1), prob=c(0.4,0.6),size = 1000, replace = TRUE)) <= 600) {
        vaild_number = vaild_number + 1
    }
    total_times = total_times + 1
}
print( vaild_number / total_times )

2

vaild_number = 0
total_times = 0
while (total_times <= 100000) {
    wt = 10 * 8 + 15 * 4
    first_time = sample(x = c(0,1), prob = c(15*4 / wt, 10*8/ wt), size = 1, replace = TRUE)
    if(first_time[1]==1){
        wt = wt - 10
        second_time = sample(x = c(0,1), prob = c(15*4/ wt, 10*7/ wt), size = 1, replace = TRUE)
    } else {
        wt = wt - 15
        second_time = sample(x = c(0,1), prob = c(15*3/ wt, 10*8/ wt), size = 1, replace = TRUE)
    }
    if(first_time[1] + second_time[1] == 2) {
        vaild_number = vaild_number + 1
    }
    total_times = total_times + 1
}
print( vaild_number / total_times )

3

random_result=rexp(10000, 0.2)
hist(random_result,freq = FALSE)
lines(density(random_result),col=14)
boxplot(random_result) 
mean(random_result)
var(random_result)
max(random_result)
min(random_result)

4

x = rnorm(10,5,9)
y = rnorm(10,5,4)
number_x = length(x)
number_y = length(y)
print((var(x)/var(y))/qf(00.975,number_x-1,number_y-1))
print((var(x)/var(y))/qf(0.025,number_x-1,number_y-1))

5

X = c(6,4,5,5,6,5,5,6,4,6,7,4)
Y = c(2,1,2,2,1,0,3,2,1,0,1,3)
alpha = 0.05
Sx = var(X)
Sy = var(Y)
nx = length(X)
ny = length(Y)
mx = mean(X)
my = mean(Y)
S_w = ((nx - 1) * Sx + (ny - 1) * Sy) / (nx + ny - 2)
t = (mx - my - 2) / S_w / sqrt(1 / nx + 1 / ny)
t >= qt(alpha, nx + ny - 2, lower.tail = FALSE)

t.test(x,y)