# What is the covariance of two independent random variables?

## What is the covariance of two independent random variables?

Finally, a covariance is zero for two independent random variables. However, a zero covariance does not imply that two random variables are independent. The magnitude of covariance depends on the variables since it is not a normalized measure.

**How do you find the covariance of two independent variables?**

Let X and Y be two independent N(0,1) random variables and Z=1+X+XY2,W=1+X. Find Cov(Z,W)….The covariance has the following properties:

- Cov(X,X)=Var(X);
- if X and Y are independent then Cov(X,Y)=0;
- Cov(X,Y)=Cov(Y,X);
- Cov(aX,Y)=aCov(X,Y);
- Cov(X+c,Y)=Cov(X,Y);
- Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z);
- more generally,

**Why is covariance of two independent variables zero?**

It’s also worthwhile to note that if two random variables are independent and thus their Covariance is 0, then the variance of their sum is just the sum of their variances (the last term – the Covariance – is 0 because the random variables are independent).

### What is the covariance of two uncorrelated variables?

Random variables whose covariance is zero are called uncorrelated. Similarly, the components of random vectors whose covariance matrix is zero in every entry outside the main diagonal are also called uncorrelated.

**Is covariance zero for independent random variables?**

If X and Y are independent variables, then their covariance is 0: Cov(X, Y ) = E(XY ) − µXµY = E(X)E(Y ) − µXµY = 0 The converse, however, is not always true.

**How do you find the covariance of two assets?**

Using our example of ABC and XYZ above, the covariance is calculated as: = [(1.1 – 1.30) x (3 – 3.74)] + [(1.7 – 1.30) x (4.2 – 3.74)] + [(2.1 – 1.30) x (4.9 – 3.74)] + … = [0.148] + [0.184] + [0.928] + [0.036] + [1.364]…Calculating Covariance.

Daily Return for Two Stocks Using the Closing Prices | ||
---|---|---|

Day | ABC Returns | XYZ Returns |

5 | 0.2% | 2.5% |

#### Does covariance equal 0 imply independence?

Zero covariance – if the two random variables are independent, the covariance will be zero. However, a covariance of zero does not necessarily mean that the variables are independent. A nonlinear relationship can exist that still would result in a covariance value of zero.

**Is covariance 0 for independent variables?**

Note that independent variables have 0 correla- tion as well as 0 covariance.

**Does uncorrelated mean zero covariance?**

Examples of dependence without correlation have zero covariance (and thus are uncorrelated), but are not independent.

## What is the covariance for a vector of random variables?

The covariance matrix is a generalization of the variance of a random variable. Remember that for a random variable, we have Var(X)=EX2−(EX)2. The following example extends this formula to random vectors. For a random vector X, show CX=RX−EXEXT.

**How do you find the covariance of two returns?**

Covariance is calculated by analyzing at-return surprises (standard deviations from the expected return) or by multiplying the correlation between the two random variables by the standard deviation of each variable.

**How do you find the covariance between two returns?**

In other words, you can calculate the covariance between two stocks by taking the sum product of the difference between the daily returns of the stock and its average return across both the stocks.

### Can independent variables be uncorrelated?

This means that independent random variables are always uncorrelated, but uncorrelated random variables may not be independent. In other words, independence is a stronger statement than uncorrelation.

**Is covariance 0 if independent?**