Trent Carpenter

2021-01-19

How can the sample covariance be used to estimate the covariance of random variables?

Usamah Prosser

Step 1
The covariance is used to measure the relationship between two random variables, that is, to what extent the variables are changing together.
When both the variables tend to move in the same direction, it is known as positive covariance.
When both the variables tend to move in the opposite direction, it is known as negative covariance.
Step 2
The formula for covariance of random variables, that is, population covariance is,
$COV\left(x,y\right)=\frac{\sum _{i=1}^{N}\left({x}_{i}-{\mu }_{x}\right)\left({y}_{i}-{\mu }_{y}\right)}{n}$
The formula for sample variance is,
$COV\left(x,y\right)=\frac{\sum _{i=1}^{N}\left({x}_{i}-\stackrel{―}{x}\right)\left({y}_{i}-\stackrel{―}{y}\right)}{n}-1$
Step 3
In the above formula for sample variance, the denominator is n – 1 which is the degrees of freedom in the sample which helps in yielding an unbiased estimator of the population. This is how the sample covariance can be used as an estimator for covariance of random variables.

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