Covariance

[SoniaKapoor]

Suppose X is a random variable, which follows Uniform[-1, 1].
Find the covariance between X and X2
(a) 1
(b) 1/4
(c) 1/8
(d) 0

[Amit Goyal]

One way to this problem is very simple. Note that U(-1, 1) is a symmetric distribution around 0. So, every odd moment will be 0 i.e. E(X) = 0, E(X^3) = 0, and so on.
Clearly, Covariance between X and X^2, Cov(X, X^2) = E(X^3) - E(X)E(X^2) = 0.

Another way to do this is by finding the integrals. Check the following file: Solution.png

[Atika Gupta]

I am getting zero as the answer. is that the answer?/

[Atika Gupta]

Basically, since X follows a uniform distribution its density function is f(x)=1/2. Therefore, E(X)=1.
X^2 follows a uniform distribution [0,1]. Hence, f(x^2)=1. hence, E(X^2)=1(u can find this expectation through integration).
Now, X^3 follows a uniform distribution [-1,1] which will again have Expectation as 1.
Thus, cov(X,Y)=E(XY)-E(X)E(Y); WHERE Y=X^2
                     =1

[Amit Goyal]

Atika, thats incorrect. Please refer to the solution I posted for the correct answer. Since X is uniform[-1, 1], E(X) = 0. X^2 take values between 0 and 1 but the distribution of X^2 is not uniform. Also, distribution of X^3 is not uniform.
Question: Find the cdf and pdfs of X^2 and X^3 when X is uniform[-1, 1].

[SoniaKapoor]

Thanku Sir :-)