a) If two random variables are independent then we can assume that their covariance is 0.
\nThe formula for covariance is:
\n\\begin{align}
Cov(X,Y) &= E[XY] - E[X]E[Y].
\\end{align}
By definition, if two random variables are independent then:
\n\\begin{align}
E[XY] &= E[X]E[Y].
\\end{align}
Hence, if we substute this back into our formula for covariance, we find
\n\\begin{align}
Cov(X,Y) &= E[XY] - E[X]E[Y] \\\\
&= E[X]E[Y] - E[X]E[Y] \\\\
&= 0.
\\end{align}
b) To calculate the covariance we can use the formula from part a), i.e.,
\n\\begin{align}
Cov(X,Y) &= E[XY] - E[X]E[Y].
\\end{align}
Hence, we must calculate $E[X]$, $E[Y]$, and $E[XY]$.
\nFirst let us caculate $E[X]$,
\n\\begin{align}
E[X] &= \\sum_{i=1}^3x_i\\cdot P(X=x_i) \\\\
&= (-1)\\cdot\\frac{1}{3} + 0\\cdot\\frac{1}{3} +1\\cdot\\frac{1}{3} \\\\
&= -\\frac{1}{3} + 0 + \\frac{1}{3} \\\\
&= 0.
\\end{align}
Since $y=X^2$, the random variable $Y$ can take the value $0$ ($Y=0^2$) or $1$ ($Y=1^2=(-1)^2$) with the probabilities $\\frac{1}{3}$ and $\\frac{2}{3}$, respectively.
\nHence,
\n\\begin{align}
E[Y] &= \\sum_{i=1}^2y_i\\cdot P(Y=y_i) \\\\
&= 0\\cdot\\frac{1}{3} + 1\\cdot\\frac{2}{3} \\\\
&= \\frac{2}{3}.
\\end{align}
Finally, we need to caluclate $E[XY]$,
\n\\begin{align}
E[XY] &= \\sum_{i=1}^3x_i\\cdot y_i\\cdot P(X=x_i,Y=y_i) \\\\
&= (-1)\\cdot (-1)^2 \\cdot \\frac{1}{3} + 0 \\cdot 0^2 \\cdot\\frac{1}{3} +1\\cdot 1^2 \\cdot \\frac{1}{3} \\\\
&= -\\frac{1}{3} + 0 + \\frac{1}{3} \\\\
&= 0.
\\end{align}
Thus,
\n\\begin{align}
Cov(X,Y) &= E[XY] - E[X]E[Y] \\\\
&= 0 - 0\\cdot \\frac{2}{3} \\\\
&= 0.
\\end{align}
c) No we cannot assume that $X$ and $Y$ are independent. Despite $Cov(X,Y)=0$ you can clearly see that $X$ and $Y$ are not independent variables since they are linearly related by $Y=X^2$.
Although two random variables being independent does mean that their covariance will be $0$, this doesn't mean that the opposite cannot be assumed true, as we have shown you can have a covariance of $0$ even when the random variables are not independent.
\n\nUse this link to find some resources which will help you revise this topic.
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We have a second random variable $Y$. Let $Y=X^2$.
\nCalculate the covariance, i.e., $Cov(X,Y)$.
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