Question

Which of these statements is false?

	If two random variables are independent, then their squares are also independent random variables
	If the joint pdf of a random vector factorises into the product of functions of the single variables, then the random variables that compose the vector are mutually independent
	If the covariance between two random variables is zero, the random variables are independent
	If two random variables are independent, their correlation is always zero
	If X is independent from Y, then the conditional pdf of X given Y is equal to the marginal pdf of X

Answer 1

(a)

If two random variables are independent then any function of those is also independent. Hence squares of those two random variables are also independent.

So, the given statement is true.

b)

For the given situation we can represent (hr) as two products as f(1,y) = f(1) * fly which occurs in case of mutually independent random variables.

So, the given statement is true.

c)

This statement is false.

Let us show this using a counter-example.

We take following set of observations.

X   Y   XY
-2   4   -8
-1   1   -1
0   0   0
1   1   1
2   4   8
Here, number of observations, n=5

Y)= (-8-1+0+1+8) = 0

(-2-1+0+1+2)=0

=(4+1+0+1+4) = 2

So, covariance is given by

Cov(X,Y) = E(XY) - E(X) EY) = 0 - 0 = 0

Clearly, here covariance between two random variables X and Y is zero. However, by observing we find a relation between X and Y as Y=X2.

So, these two random variables are not at all independent though covariance is zero.

Note- Covariance between two random variables being zero can imply that those are linearly independent. Non-linear dependence or independence can not be judged directly using covariance value only. That is why we constructed the counter example using a non-linear (quadratic) case.

So, the given statement is false.

d)

If two random variables are independent then E(XY) = E(X) E(Y) and so

Cov(X,Y) = E(XY) - E(X)E(Y) = E(X) E(Y) - E(X)E(Y) = 0.

As covariance term is in the numerator of the formula of correlation coefficient, so the correlation coefficient is also zero.

So, the given statement is true.

e)

From the definition of conditional probability,

P(X | Y) - P(X n Y) /P(Y)

For the case of independence,

P( X n Y ) = P(X)P(Y)

Using this result in the previous one we get,

P(X | Y) =p(X)P(Y)/P(Y)=P(X)

So, the given statement is true.

The first statement is false.