Question

Let \(X = \cos\Theta\) and \(Y = \sin\Theta\), where \(\Theta\) is a random variable uniformly distributed over \((0, 2\pi)\). Which of the following options is/are CORRECT?

Hint: \( \text{cov}(X,Y) = E\big[(X-E(X))(Y-E(Y))\big] \)

• A. \(X\) and \(Y\) are dependent and uncorrelated
• B. \(X\) and \(Y\) are independent and uncorrelated
• C. \(X\) and \(Y\) are dependent and correlated
• D. \(X\) and \(Y\) are independent and correlated

Hint:

Check covariance using the hint formula, then separately check whether X and Y satisfy a deterministic relationship such as \(X^2+Y^2=1\).

Answer 1

The Correct Option is A

Step 1: Argue cov(X,Y)=0 using symmetry instead of direct integration.
Consider the substitution $\theta' = -\theta$, equivalently $\theta' = 2\pi - \theta$, which maps the uniform distribution on $(0,2\pi)$ back onto itself, so $\theta'$ has exactly the same distribution as $\theta$. Under this substitution, $\cos\theta' = \cos\theta$, X is unchanged, while $\sin\theta' = -\sin\theta$, Y flips sign. Therefore the product $XY = \cos\theta\sin\theta$ maps to $-XY$ under a transformation that leaves the distribution unchanged, which forces \[ E[XY] = E[-XY] \implies 2E[XY] = 0 \implies E[XY] = 0 \] Combined with $E[X]=E[Y]=0$, each is the average of a sinusoid over a full period, zero by symmetry, the hint gives $cov(X,Y) = E[XY]-E[X]E[Y] = 0$. So X and Y are uncorrelated.

Step 2: Argue dependence using a purely geometric picture.
The pair $(X,Y) = (\cos\theta, \sin\theta)$ is just the point where the ray at angle $\theta$ meets the unit circle $x^2+y^2=1$. Every possible outcome of the random experiment lies exactly on this circle, never anywhere else in the plane. If X and Y were independent, fixing X should leave Y free to range over its entire marginal support regardless of the value of X, but fixing $X = 1/2$ forces the point onto the circle at only two possible spots, $Y = +\frac{\sqrt3}{2}$ or $Y = -\frac{\sqrt3}{2}$, and knowing X = 1 forces Y = 0 exactly. Since the conditional behavior of Y changes sharply with the value of X, the two variables cannot be independent.

Step 3: Combine both results.
Zero covariance, Step 1, shows there is no linear association between X and Y, but the circular constraint, Step 2, shows there is a strong nonlinear, in fact deterministic up to a sign, relationship between them. Zero correlation captures only linear dependence, so this is a classic example where uncorrelated does not imply independent.

Step 4: Select the matching option.
X and Y are dependent, Step 2, and uncorrelated, Step 1, which is option (A). This rules out (B) independent and uncorrelated, and (C), (D) which both claim correlation exists.

\[ \boxed{\text{Option (A) is correct}} \]