1. **Problem statement:** Show that $\mathrm{Cov}[aX, Y] = a \mathrm{Cov}[X, Y]$ where $a \in \mathbb{R}$. 2. **Recall the definition of covariance:** $$\mathrm{Cov}[X, Y] = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]$$ 3. **Apply the definition to $\mathrm{Cov}[aX, Y]$:** $$\mathrm{Cov}[aX, Y] = E[(aX - E[aX])(Y - E[Y])]$$ 4. **Use linearity of expectation:** Since $E[aX] = aE[X]$, rewrite as $$\mathrm{Cov}[aX, Y] = E[(aX - aE[X])(Y - E[Y])] = E[a(X - E[X])(Y - E[Y])]$$ 5. **Factor out the constant $a$ from expectation:** $$\mathrm{Cov}[aX, Y] = a E[(X - E[X])(Y - E[Y])] = a \mathrm{Cov}[X, Y]$$ 6. **Conclusion:** We have shown that multiplying the random variable $X$ by a scalar $a$ scales the covariance by the same factor $a$. Thus, $$\boxed{\mathrm{Cov}[aX, Y] = a \mathrm{Cov}[X, Y]}$$