The covariance of a sum of two random variables X and Y

[Ad VanderVen]

TL;DR Summary

Suppose you have three random variables X, Y and K. Suppose X and Y are independent, but each correlated with K. Suppose Z = X+Y. Is it true that in probability theory the covariance of Z with K is equal to the sum of the covariance of X with K and the covariance of Y with K?

Suppose X and Y are random variables. Is it true that

Cov (Z,K) = Cov(X,K)+Cov(Y,K) where Z=X+Y?

 

[Ad VanderVen]

The summary:

Suppose you have three random variables X, Y and K. Suppose X and Y are independent, but each correlated with K. Suppose Z = X+Y. Is it true that in probability theory the covariance of Z with K is equal to the sum of the covariance of X with K and the covariance of Y with K?

is incorect and should be:

Suppose you have three random variables X, Y and K. Suppose Z = X+Y. Is it true that in probability theory the covariance of Z with K is equal to the sum of the covariance of X with K and the covariance of Y with K?

 

[FactChecker]

Covariance is linear in each variable. The random variables do not have to be independent. See this.

 

[Ad VanderVen]

Covariance is linear in each variable. The random variables do not have to be independent. See this.

The answer indeed follows from Wikipedia: article Covariance, section Covariance of linear combinations (sixth equation).

Reference: https://www.physicsforums.com/threads/the-covariance-a-sum-of-two-random-variables-x-and-y.1056159/

 

[WWGD]

I remember something along the lines that correlation( covariance?) was an inner -product in some space of Random Variables. I guess we have Cov<X,X>=Var(X)=norm(X)?

 

[BWV]

How can X and Y be independent (cov=0) if they are both correlated with K?

 

[FactChecker]

How can X and Y be independent (cov=0) if they are both correlated with K?

Suppose X and Y are any two independent variables and K = X+Y. Then cov(X,K) = cov(X, X+Y) = cov(X,X) + cov(X,Y) = cov(X,X) + 0 > 0.

 

[BWV]

Let X and Y be independent uniformly distributed RVs and let K be some function of them.

Z=X+Y

But k could be anything, if it is X+Y it obviously perfectly correlated, but is could also be X-Y, or sin(x+y) etc, no?

 

[WWGD]

I believe the best we can use is bilinearity of coefficients, i.e.,
Cov(aX, bY)=abCov(X,Y).
But you're right, beyond that, I think there are no rules for f with f=f(X,Y).

Edit: I suspect the answer here may fall under propagation of errors/uncertainty

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Propagation of uncertainty​

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For the propagation of uncertainty through time, see Chaos theory § Sensitivity to initial conditions.
In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate due to the combination of variables in the function.
The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, which is the positive square root of the variance. The value of a quantity and its error are then expressed as an interval x ± u. However, the most general way of characterizing uncertainty is by specifying its probability distribution. If the probability distribution of the variable is known or can be assumed, in theory it is possible to get any of its statistics. In particular, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are approximately ± one standard deviation σ from the central value x, which means that the region x ± σ will cover the true value in roughly 68% of cases.
If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.[1]
In a general context where a nonlinear function modifies the uncertain parameters (correlated or not), the standard tools to propagate uncertainty, and infer resulting quantity probability distribution/statistics, are sampling techniques from the Monte Carlo method family.[2] For very expansive data or complex functions, the calculation of the error propagation may be very expansive so that a surrogate model[3] or a parallel computing strategy[4][5][6] may be necessary.
In some particular cases, the uncertainty propagation calculation can be done through simplistic algebraic procedures. Some of these scenarios are described below.

Linear combinations​

Non-linear combinations​

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Example formulae​

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Example calculations​

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See also​

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Further reading​

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Last edited 1 month ago by Hellacioussatyr

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[FactChecker]

Let X and Y be independent uniformly distributed RVs and let K be some function of them.

Z=X+Y

But k could be anything,

Well, it can be a lot of things. If you start with any two uncorrelated variables, X and Y, many variables, Z, will have some correlation with both of them.

if it is X+Y it obviously perfectly correlated,

Not perfectly. X's correlation with X+Y is not perfect. The independent Y variable term prevents that.

but is could also be X-Y, or sin(x+y) etc, no?

Yes, there are a lot of examples where there is clearly a connection. There are also a lot of examples where two uncorrelated variables, X and Y might be correlated with a third variable, K, with no apparent reason.

 

[BWV]

Well, it can be a lot of things. If you start with any two uncorrelated variables, X and Y, many variables, Z, will have some correlation with both of them.

Not perfectly. X's correlation with X+Y is not perfect. The independent Y variable term prevents that.

Yes, there are a lot of examples where there is clearly a connection. There are also a lot of examples where two uncorrelated variables, X and Y might be correlated with a third variable, K, with no apparent reason.

Was thinking of the correlation of z and k per the OP, obviously if Z=K the correlation is 1