Authorities say nobody on the ground was hurt.
The pilot, who is male, was the only person on board.
But his condition is not yet known.
It is also understood that he was preparing for an air show.
Television news footage shows debris spread across a scorched patch of empty yard.
An investigation will be launched by the Federal Aviation Administration (FAA) and the National Transportation Safety Board.
The prison in Riverside County has 3,400 male inmates.
The incident happened on the same day as a plane crash in Texas.
A twin-engine Beech BE58 aircraft crashed into a ravine around 9:00am local time (4:00pm BST) and killed all six passengers on board, the Texas Department of Public Safety said.
It departed from the West Houston Airport and crashed in Kerrville.
The plane was scheduled to land at Kerrville Municipal Airport.
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