Calculating the variance of a random variable

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I have stumbled upon this question from my text book and I have been finding difficulty in understanding and solving it.

If X and Y are independent random variables with variances $σ_X^2 = 5$ and $σ_Y^2 = 3$. Find the variance of the random variable $Z = −2X + 4Y − 3$. Repeat afterwards this exercise, with $X$ and $Y$ not independent and $σ_{XY} = 1$.

Far as I know, if you have independent variables then $\textrm{var}(X+Y)= \textrm{var}(X)+\textrm{var}(Y)$ also $\textrm{var}(XY)=\textrm{var}(X)\textrm{var}(Y)$.

For the first question I just filled in 5 and 3:

$$Z=-2(5)+4(3)-3=-1$$

For the second question, I have absolutely no idea how to approach it.. Any help would be appreciated

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1 Answer

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Let's do this from the definitions. Note that$$ Var X = \mathbb{E}\left[X^2\right] - \mathbb{E}[X]^2 = m_{X^2} - m_X^2, $$with $m$ denoting the expected value for a quick shorthand.

Therefore,$$ \begin{split} Var (X+Y) &= \mathbb{E}\left[(X+Y)^2\right] - \mathbb{E}[X+Y]^2 \\ &= \mathbb{E}\left[X^2\right] + \mathbb{E}\left[Y^2\right] + 2\mathbb{E}\left[XY\right] - m_X^2 - m_Y^2 - 2m_Xm_Y \\ &= \sigma_X^2 + \sigma_Y^2 + 2(\mathbb{E}[XY] - m_X m_Y) \end{split} $$Now, if $X,Y$ are independent,$$ \begin{split} \mathbb{E}[XY] &= \iint_\mathbb{R^2} xy f_{X,Y}(x,y)dxdy \\ &= \iint_\mathbb{R^2} xy f_X(x) f_Y(y) dxdy \\ &= \left(\int_\mathbb{R} xf_X(x) \right)\left(\int_\mathbb{R} yf_Y(y) \right) \\ &= m_X m_Y. \end{split} $$This implies $Var(X+Y) = \sigma_X^2 + \sigma_Y^2$. When $X,Y$ are dependent, that's not necessarily true anymore.

In particular, if you let $X=Y$, it's easy to prove that$$ Var(X+Y) = Var(X+X) = Var(2X) = 4\sigma_X^2. $$

Can you finish this?

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