In the DTLS handshake, the two parties involved in the communication must authenticate each other by means of certificates, and after that they perform an ephemeral Diffie-Hellman key exchange, with the public Diffie-Hellman keys signed with the keys of the certificates.
Why EDH is needed in this context? If each party knows the other party's public key, why can't one party chose a random key and send it to the other one? What kind of security adds EDH in this context?