$\mathbf{5 . 2} 020$ balls are placed into 2020 different boxes, with each ball having an equal probability of landing in any box, and the probabilities of different balls landing in the boxes are independent of each other. The expected number of empty boxes is $\qquad$
5. $\frac{2019^{2020}}{2020^{2019}}$.

Define the random variable $x_{i}$: if the $i$-th box has a ball, then $x_{i}=0$, otherwise, $x_{i}=1$.
Thus, $P\left(x_{i}=1\right)=\frac{2019^{2020}}{2020^{2020}}$,
$P\left(x_{i}=0\right)=1-\frac{2019^{2020}}{2020^{2020}}$.
Therefore, $\mathrm{E}\left(x_{i}\right)=1 \times \frac{2019^{2020}}{2020^{2020}}=\frac{2019^{2020}}{2020^{2020}}$.
Let the random variable for the number of empty boxes be $\xi=\sum_{i=1}^{2020} x_{i}$.
Thus, $\mathrm{E}(\xi)=\mathrm{E}\left(\sum_{i=1}^{2020} x_{i}\right)=\sum_{i=1}^{2020} \mathrm{E}\left(x_{i}\right)$
$=2020 \times \frac{2019^{2020}}{2020^{2020}}=\frac{2019^{2020}}{2020^{2019}}$.