Do all musicians write their own songs?

If a singer is very talented but can't write their own songs, they hire songwriters who can't really sing to do it for them. Or, if they just need a little help, they co-write with a professional songwriter where the musician comes in with the ideas and the songwriter helps them make it into a wonderful song. But if a singer is an Excellent songwriter also, they write their own songs.

Singers who write their own songs:

Taylor Swift

Lady Gaga

Prince

Rhianna

Beyonce

Jonas brothers

Natasha Bedingfield

Britney Spears

Katy Perry

Singers who don't:

Miley Cyrus

Demi Lovato

Selena Gomez

Miranda Cosgrove