Can you construct a pair of random variables such that
Var(X)=Var(Y)=1 and Cov(X,Y)=2?
No.
Corr(X,Y) = Cov(X,Y)/sqrt(Var(X)Var(Y)).
If Var(X) = Var(Y) = 1 and Cov(X,Y) = 2, then we have Corr(X,Y) = 2/sqrt(1*1) = 2.
However, Corr(X,Y) scales from -1 to 1. Contradiction.
r45=4562
Sure, let's have some fun constructing these random variables!
Let's say we have two random variables, X and Y.
To start off, let's make X to be a standard normal distribution, which has a variance of 1.
Now, to make things interesting, let's make Y to be X plus 2 times a standard normal distribution. This means, Y = X + 2Z, where Z is also a standard normal distribution.
Now, let's calculate the variances and covariance:
Var(X) = Var(Y) = Var(X + 2Z)
= Var(X) + 2 * Cov(X,Z) + 4 * Var(Z)
= Var(X) + 2 * 0 + 4 * 1 (since X and Z are independent)
= 1 + 0 + 4
= 5
Oops! Looks like we made a silly mistake along the way. Our original intention was to have Var(X) = Var(Y) = 1, but it seems like we went off track.
Well, such is the nature of random things - sometimes they don't work out as planned. But hey, at least we had a good laugh along the way, right?
To construct a pair of random variables with Var(X) = Var(Y) = 1 and Cov(X, Y) = 2, we need to consider the relationship between the variances and the covariance.
The covariance between two random variables X and Y is defined as Cov(X, Y) = E[(X - μx)(Y - μy)], where E denotes the expected value and μx and μy represent the means of X and Y, respectively.
To achieve Cov(X, Y) = 2, we need to find two random variables that have a positive correlation.
Let's consider the following construction:
1. Let X be a standard normal random variable with mean 0 and variance 1.
2. Let Y = X + 2.
Here, both X and Y have a variance of 1 since Var(X) = Var(Y) = 1.
Now, let's calculate the covariance:
Cov(X, Y) = E[(X - μx)(Y - μy)]
= E[X(X + 2)]
= E[(X^2 + 2X)]
= E[X^2] + 2E[X]
E[X] = μx = 0 since X is a standard normal random variable with mean 0.
E[X^2] = Var(X) + (E[X])^2 = 1 + 0^2 = 1.
Thus, Cov(X, Y) = E[X^2] + 2E[X] = 1 + 2(0) = 1.
So, the covariance between X and Y is 1, not 2.
Unfortunately, it is not possible to construct two random variables with Var(X) = Var(Y) = 1 and Cov(X, Y) = 2.
To construct a pair of random variables X and Y such that Var(X) = Var(Y) = 1 and Cov(X, Y) = 2, we need to consider the definition of covariance and use it to find the correlation coefficient.
The covariance between two random variables X and Y is defined as:
Cov(X, Y) = E[(X - E[X])(Y - E[Y])]
where E[.] denotes the expected value.
Now, to find the correlation coefficient, we divide the covariance by the square root of the product of the variances:
Corr(X, Y) = Cov(X, Y) / sqrt(Var(X) * Var(Y))
In this case, we know that Var(X) = Var(Y) = 1 and Cov(X, Y) = 2. Let's substitute these values into the formula:
Corr(X, Y) = 2 / sqrt(1 * 1)
= 2 / 1
= 2
Now, the correlation coefficient lies between -1 and 1. Since the correlation coefficient we obtained is 2, which is greater than 1, it is not possible to find random variables satisfying these conditions.
Therefore, it is not possible to construct a pair of random variables X and Y such that Var(X) = Var(Y) = 1 and Cov(X, Y) = 2.