Joseph married Cornelia Curtis Jennings 21 May 1868. Cornelia (daughter of Charles Pierson Jennings and Gertrude Gouverneur Burnet) was born 8 May 1845, Michigan; died 15 Aug 1926, Shelbyville, Shelby County, Indiana. [Group Sheet]

1. Albert Vernon Randall was born Apr 1869, Shelbyville, Shelby County, Indiana; died 13 Nov 1925, Shelby County, Indiana; was buried , Shelbyville, Shelby County, Indiana.
2. Burnet Scott Randall was born 9 Nov 1870, Selby County, Indiana; died 10 Aug 1877, Selby County, Indiana.
3. Walter Shellman Randall was born 3 Jan 1872, Shelbyville, Indiana; died 1941, Sarasota, Florida, United States.
4. Gertrude Randall was born 15 Nov 1875, Shelbyville, Indiana; died 31 Mar 1974, Indianapolis, Marion County, Indiana.
5. Bertha Randall was born 24 Jun 1878, Shelbyville, Indiana; died 6 Sep 1973, Indianapolis, Marion County Indiana; was buried , Shelbyville, Shelby County, Indiana.