Question

Using R programming language, complete the following.

1. Generate the bivariate normal sample of size 100 with parameters
a. marginal mean of X equal to 1,
b. marginal mean of Y equal to 2,
c. marginal variance of X equal to 3,
d. marginal variance of Y equal to 4,
e. correlation between X and Y equal to -1/2.
You can use the function mvrnorm(), first installing its package by the code
if (! require ("MASS")) install.packages("MASS"); library ("MASS")

2. Produce scatter plot.

3. Create histogram of a marginal distribution of X. You can use the function hist()

4. Create qq plot (probability plot) of a marginal distribution of X vs normal and comment on it. You can use the function qqnorm().

5. Compute sample means and sample variances of X and Y marginal distributions, and sample correlation between X and Y. You can use the functions mean(), var(), cor().

Answer 1

3.

4.

As from QQ plot we can see that the points follow linear line so we can say that X and Y come from normal distribution.

(5)
[1] 1.245655
[1] 2.053158
> var(X);var(Y)
[1] 3.048119
[1] 3.305575
> cor(X,Y)
[1] -0.09814289

The code that we used are below.

library(MASS)
mu=c(1,2)
Sigma=matrix(c(3,-.5,-.5,4),2,2)
xx=mvrnorm(n = 100, mu, Sigma, tol = 1e-6, empirical = FALSE, EISPACK = FALSE)
X=xx[,1];Y=xx[,2]
plot(x,xlab = "X",ylab = "Y")
hist(X)
hist(Y)

qqnorm(X,main = "Normal QQ Plot of X")
qqnorm(Y,main = "Normal QQ Plot of Y")

mean(X);mean(Y)
var(X);var(Y)
cor(X,Y)