Let { N ( t ) , t > = 0 } be a non - homogeneous Poisson process with intensity function \ lambda ( t ) = { 2 + t , if 0 < = t < = 2 , 4 , otherwise. ( a ) Compute P { N ( 3 ) = 7 | N ( 1 ) = 5 } . ( b ) Compute E [ N ( 4 ) N ( 1 ) | N ( 3 ) = 2 ] .