If cov(X,Y)=15, what is restrictions should we put standard deviation of X and y?

Question

Answer

Restrictions on Standard Deviation of X and Y given Covariance

When we have the covariance between two variables X and Y, we can determine some restrictions on their standard deviation. The covariance between X and Y is given by:

Cov(X,Y) = E[(X - E[X])(Y - E[Y])]

where E[X] and E[Y] are the expected values of X and Y, respectively.

Positive Covariance

If the covariance between X and Y is positive, i.e., Cov(X,Y) > 0, then:

Both X and Y tend to increase or decrease together.

The standard deviation of X and Y must be positive.

The standard deviation of X and Y cannot be zero.

The standard deviation of X and Y can be anything greater than zero.

Negative Covariance

If the covariance between X and Y is negative, i.e., Cov(X,Y) < 0,='' />

X tends to increase when Y decreases and vice versa.

The standard deviation of X and Y must be positive.

The standard deviation of X and Y cannot be zero.

The standard deviation of X and Y can be anything greater than zero.

Zero Covariance

If the covariance between X and Y is zero, i.e., Cov(X,Y) = 0, then:

X and Y are independent of each other.

The standard deviation of X and Y can be anything greater than or equal to zero.

If the standard deviation of X or Y is zero, then the variable is constant and not informative.

Conclusion

In conclusion, the covariance between X and Y provides information about their linear relationship. The standard deviation of X and Y must be positive and cannot be zero. If the covariance is zero, then the variables are independent, and their standard deviations can be anything greater than or equal to zero.

Answer

The product of standard deviations must be greater than covariance i.e. 15

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