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Phenylenes belong to a special class of conjugated hydrocarbons composed of a special arrangement of hexagons and squares. In the phenylene structure, every square is adjacent to a pair of hexagons, and any two hexagons are not adjacent. If in the structure of phenylene only two squares are adjacent to every hexagon, then it is called a phenylene chain. In this paper, we compute the expected values of the first and second Gourava index, the first and second redefined Zagreb index, and the Hyper-Zagreb index of random phenylene chains. Furthermore, a comparison between the expected values of the computed indices among the classes of random phenylene chains has been investigated. [ABSTRACT FROM AUTHOR]