3) Association between Random Variables¶
Vitor Kamada
econometrics.methods@gmail.com
Last updated 7-18-2020
3.1) If $6000 is invested in Amazon, and $2000 in Intel, what is the expected return?¶
Suppose Amazon and Intel stocks have recently been rising 2.1% and 0.4% per month, respectively.
Let’s \(Z\) be the portfolio composed by stocks X and Y:
Then:
$$ =$134 $$
3.2) What is the variance of the portfolio Z?¶
Based on previous question:
If X and Y are independent:
Suppose that the variance of Amazon and Intel stocks are respectively 0.72% and 0.27% per month.
This value is hard to interpret, because the measurement unit is a square term.
3.3) What is the standard deviation of the portfolio Z?¶
$$=\sqrt{$^2270,000}$$
$$ = $520 $$
An outcome in the range of $520 above or below the expected return is very likely to occur.
3.4) What is the covariance between Random Variables?¶
The covariance between random variables is the expected value of the product of deviations from the means.
It is hard to interpret, because the units of random variables X and Y are multiplied.
3.5) What is \(Cov(X,X)\)?¶
By definition:
Then:
Therefore, variance is a special case of covariance.
3.6) Show that \(Cov(X,Y)=E(XY)-\mu_x\mu_y\).¶
By definition:
Then:
3.7) Prove that if \(X\) and \(Y\) are independent, then \(Cov(X,Y)=0\).¶
Remember the definition of two independent events A and B:
If \(X\) and \(Y\) are independent random variables, then:
From previous question:
Intuition: If X and Y are independent random variables, knowledge or information of random variable X doesn’t help in predicting the outcome of random variable Y, and vice-versa. Therefore, there is no linear association between these two variables: \(Cov(X,Y)=0\).
3.8) Show that \(Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)\).¶
By definition:
Then:
3.9) What is correlation, \(\rho\)?¶
The correlation is the covariance divided by the product of standard deviations:
The correlation is easy to interpret. It ranges from -1 to +1.
It is unit free. The units from covariance are removed by the division of the product of standard deviations.
3.10) If \(X\) and \(Y\) are uncorrelated (\(\rho =0\)), it is possible to conclude that \(X\) and \(Y\) are independent?¶
Suppose that the joint probability distribution, \(P(X,Y)\), is given by the table below.
The random variables X and Y are independent if:
\(P(X=10, Y=0)=1/2\)
\(P(X=10) = 0+\frac{1}{2} +0\)
\(P(Y=0) = \frac{1}{2} + 0\)
Therefore, X and Y are dependent:
Let’s convert the table in a format useful for calculations:
x = [10, 10, 10, -10, -10, -10]
y = [-10, 0, 10, -10, 0, 10]
probXY = [0, 1/2, 0, 1/4, 0, 1/4]
dictionary = {'x': x, 'y': y, 'P':probXY }
import pandas as pd
table = pd.DataFrame(dictionary)
table
| x | y | P | |
|---|---|---|---|
| 0 | 10 | -10 | 0.00 |
| 1 | 10 | 0 | 0.50 |
| 2 | 10 | 10 | 0.00 |
| 3 | -10 | -10 | 0.25 |
| 4 | -10 | 0 | 0.00 |
| 5 | -10 | 10 | 0.25 |
We calculated that \(\mu_x=0\), and \(\mu_y=0\).
table['x*P'] = table['x']*table['P']
table['y*P'] = table['y']*table['P']
table
| x | y | P | x*P | y*P | |
|---|---|---|---|---|---|
| 0 | 10 | -10 | 0.00 | 0.0 | -0.0 |
| 1 | 10 | 0 | 0.50 | 5.0 | 0.0 |
| 2 | 10 | 10 | 0.00 | 0.0 | 0.0 |
| 3 | -10 | -10 | 0.25 | -2.5 | -2.5 |
| 4 | -10 | 0 | 0.00 | -0.0 | 0.0 |
| 5 | -10 | 10 | 0.25 | -2.5 | 2.5 |
sum(table['x*P'])
0.0
sum(table['y*P'])
0.0
We get that \(E(XY) =0\).
table['x*y'] = table['x']*table['y']
table['x*y*P'] = table['x*y']*table['P']
table
| x | y | P | x*P | y*P | x*y | x*y*P | |
|---|---|---|---|---|---|---|---|
| 0 | 10 | -10 | 0.00 | 0.0 | -0.0 | -100 | -0.0 |
| 1 | 10 | 0 | 0.50 | 5.0 | 0.0 | 0 | 0.0 |
| 2 | 10 | 10 | 0.00 | 0.0 | 0.0 | 100 | 0.0 |
| 3 | -10 | -10 | 0.25 | -2.5 | -2.5 | 100 | 25.0 |
| 4 | -10 | 0 | 0.00 | -0.0 | 0.0 | 0 | 0.0 |
| 5 | -10 | 10 | 0.25 | -2.5 | 2.5 | -100 | -25.0 |
sum(table['x*y*P'])
0.0
Therefore: \(Cov(X,Y)=E(XY)-\mu_x\mu_y = 0\)
We cannot conclude that if correlation is zero, the two random variables are independent.
Intuition: Dependency is a stronger concept than correlation. Dependency capture any relationship between variables, ex: quadratic relationship; whereas, correlation only capture linear relationship.
Exercises¶
1| If you want a portfolio with small risk (variance), should you look for investments that have positive covariance, have negative covariance, or are uncorrelated? Justify.
2| When \(Var(X+Y)=Var(X)+Var(Y)\)? Explain.
3| Does a portfolio formed from the mix of three investments have more risk (variance) than a portfolio formed from two? Make assumptions and justify you reasoning.
4| What is the covariance between a random variable and itself?
5| What is the correlation between a random variable and itself?
6| What’s the covariance between a random variable X and a constant?
7| Suppose a random variable \(X\) can take only the three values -1, 0, and 1, and that each of these three values has the same probability. Let \(Y = X^2\).
a) Show that \(Correlation(X,Y) = 0\).
b) Are X and Y independent? Justify.