  We prove that a random word of length $n$ over a $k$-ary fixed alphabet
contains, on expectation, $\Theta(\sqrt{n})$ distinct palindromic factors. We
study this number of factors, $E(n,k)$, in detail, showing that the limit
$\lim_{n\to\infty}E(n,k)/\sqrt{n}$ does not exist for any $k\ge2$,
$\liminf_{n\to\infty}E(n,k)/\sqrt{n}=\Theta(1)$, and
$\limsup_{n\to\infty}E(n,k)/\sqrt{n}=\Theta(\sqrt{k})$. Such a complicated
behaviour stems from the asymmetry between the palindromes of even and odd
length. We show that a similar, but much simpler, result on the expected number
of squares in random words holds. We also provide some experimental data on the
number of palindromic factors in random words.
