Calculating Expected Values

Given $X = \cos \theta$ and $Y = \sin \theta$, where $\theta$ is uniformly distributed over $(0, 2\pi)$. The probability density function (PDF) of $\theta$ is $f(\theta) = \frac{1}{2\pi}$ for $0 < \theta < 2\pi$.

• $E(X) = E(\cos \theta) = \int_{0}^{2\pi} \cos \theta \cdot \frac{1}{2\pi} d\theta = \frac{1}{2\pi} [\sin \theta]_{0}^{2\pi} = 0$.
• $E(Y) = E(\sin \theta) = \int_{0}^{2\pi} \sin \theta \cdot \frac{1}{2\pi} d\theta = \frac{1}{2\pi} [-\cos \theta]_{0}^{2\pi} = 0$.

Determining Correlation using Covariance

The hint provides the formula for covariance: $\text{cov}(X, Y) = E[(X - E(X)) (Y - E(Y))]$. This simplifies to $\text{cov}(X, Y) = E(XY) - E(X)E(Y)$.

• Since $E(X) = 0$ and $E(Y) = 0$, the covariance simplifies to $\text{cov}(X, Y) = E(XY)$.
• Calculate $E(XY)$: $XY = \cos \theta \sin \theta$. Using the double angle identity, $XY = \frac{1}{2} \sin(2\theta)$.
• $E(XY) = E\left(\frac{1}{2} \sin(2\theta)\right) = \int_{0}^{2\pi} \frac{1}{2} \sin(2\theta) \cdot \frac{1}{2\pi} d\theta$
• $E(XY) = \frac{1}{4\pi} \int_{0}^{2\pi} \sin(2\theta) d\theta = \frac{1}{4\pi} \left[-\frac{1}{2} \cos(2\theta)\right]_{0}^{2\pi} = \frac{1}{4\pi} \left(-\frac{1}{2} \cos(4\pi) - (-\frac{1}{2} \cos(0))\right) = \frac{1}{4\pi} \left(-\frac{1}{2} - (-\frac{1}{2})\right) = 0$.
• Therefore, $\text{cov}(X, Y) = 0$. This means $X$ and $Y$ are uncorrelated.

Assessing Independence

Two random variables are independent if the occurrence of one does not affect the probability of the other.

• Consider the fundamental trigonometric identity: $X^2 + Y^2 = (\cos \theta)^2 + (\sin \theta)^2 = 1$.
• This equation shows a direct functional relationship between $X$ and $Y$. If the value of $X$ is known, the possible values for $Y$ are restricted (e.g., if $X=1$, $Y$ must be $0$).
• Because a functional relationship exists, $X$ and $Y$ cannot be independent.

Conclusion

The analysis shows that:

• $X$ and $Y$ are uncorrelated because their covariance is $0$.
• $X$ and $Y$ are dependent due to the relationship $X^2 + Y^2 = 1$.

Therefore, the correct option is that $X$ and $Y$ are dependent and uncorrelated.