Analyzing the covariance and correlation between two independent standard Gaussian random variables

Question 3
Analytical
Solution
Given X and Y are independent standard Gaussian r.v’s
Step 1:
A Gaussian distribution is such that
X~ N (0, 1), Y~N (0, 1) with PDF:
FXX= 1
√ 2 π e{− x2
x }
Fy Y= 1
√2 π e{− y2
y }
Since the Gaussian is a normally distributed, therefore, it has mean 0 and variance 1, i.e. normal
distributions have mean 0 and variance 1
Hence:
Step 2:
Calculating covariance
Covariance
Covariance of X and Y is given as

Cov (X, Y) = E [XY] - E[X].E[Y]
But given that the sample mean is calculated as:
E[Y]= y= 1
2 ∑
i=1
2
(2+3)=2.500
E[X]=x= 1
2 ∑
i=1
2
(1−1)=0.000
Therefore:
Step 3:
Sample covariance is calculated as in:
𝓸xy = 1
n−1 ∑
i=1
n
(xi−¿ x )( yi− y )¿
Hence from values from the distribution and sample means of both x and y
1
2−1 ∑
1
2
(1−0.000)(2−2.5)+ 1
2−1 ∑
1
2
(−1−0.000)(3−2.500)= -1
Thus cov of x and y= -1
*This is true Since independent random variables are not correlated, their covariance ranges
between (-1,0,1)