# i)
x <- rep(1, 100000)
for (i in 1:100000) {
x[i] = (sum(runif(3)) - 3/2)/(sqrt(3/12))
}
qqnorm(x)
mean(x)
sd(x)
# For 95% CI data should lies between
qnorm(0.025)
# and
qnorm(0.975)
#now from the data we need to see the proportion of observations in this interval
length(x[x < 1> -1.959964])/100000
# So proportion of data between given range is given above
# Similarly for 0.5 and 99.5 percentile is
length(x[x < qnorm> qnorm(0.005)])/100000
# now for (ii)
x1 <- rep(1, 100000)
for (i in 1:100000) {
x1[i] = qnorm(runif(1))
}
qqnorm(x1)
mean(x1)
sd(x1)
# For 95% CI data should lies between
qnorm(0.025)
# and
qnorm(0.975)
#now from the data we need to see the proportion of observations in this interval
length(x1[x1 < 1> -1.959964])/100000
# So proportion of data between given range is given above
# Similarly for 0.5 and 99.5 percentile is
length(x1[x1 < qnorm> qnorm(0.005)])/100000
# So as we can see that both the distribution are following normal distribution
# Q 2
x <- rnorm(1000)
y <- x^2
#A)
# So here we will find the probability from sample in such a way that number of observation less than 1.2 divided by total number of observations
length(y[y<=1.2])/1000
#B)
pchisq(1.2, 1)
#C)
# There is a difference between both of them because one of them is determining the probabilty based on the random sample drawn. And another one is determining the exact probability.
#D)
#Estimate of m is given by :
mean(y)
#E)
# As we know that here m = 1 accorording to theoritical distribution
# So if we increase N then value of m will close to 1