Although the output of your query looks rather simple, it may be an advantage to write an inline view for generating values that can be utilised in an outer SELECT eg (using your example table)
Table and data
create table example ( id, day_ )
as
select 1, date '2020-02-02' from dual union all -- 2-2-20
select 2, date '2020-01-03' from dual union all -- 3-1-20
select 3, date '2020-02-15' from dual union all -- 15-2-20
select 4, date '2020-03-20' from dual -- 20-3-20
;
First step ( use this for the "inline view")
select
to_char( day_, 'MM-YY') month_
, to_char( extract ( day from last_day( day_ ) ) ) lastday_
, count(*) over ( order by to_char( day_, 'MM-YY') ) runningtotal_
, row_number() over ( partition by to_char( day_, 'MM-YY') order by day_ ) rn_
from example ;
-- result
+------+--------+-------------+---+
|MONTH_|LASTDAY_|RUNNINGTOTAL_|RN_|
+------+--------+-------------+---+
|01-20 |31 |1 |1 |
|02-20 |29 |3 |1 |
|02-20 |29 |3 |2 |
|03-20 |31 |4 |1 |
+------+--------+-------------+---+
Final query
select
month_
, runningtotal_ rows_num
, round( max( rn_ ) / lastday_, 5 ) avg_per_day
, to_char( max( rn_ ) ) || '/' || to_char( lastday_ ) avg_per_day
from (
select
to_char( day_, 'MM-YY') month_
, to_char( extract ( day from last_day( day_ ) ) ) lastday_
, count(*) over ( order by to_char( day_, 'MM-YY') ) runningtotal_
, row_number() over ( partition by to_char( day_, 'MM-YY') order by day_ ) rn_
from example
)
group by month_, runningtotal_, lastday_
order by month_
;
-- result
+------+--------+-----------+-----------+
|MONTH_|ROWS_NUM|AVG_PER_DAY|AVG_PER_DAY|
+------+--------+-----------+-----------+
|01-20 |1 |0.03226 |1/31 |
|02-20 |3 |0.06897 |2/29 |
|03-20 |4 |0.03226 |1/31 |
+------+--------+-----------+-----------+
DBfiddle here.
Note: It was not quite clear (to me) which form of "AVG_PER_DAY" you require. Just delete the line that you do not need from the outer SELECT.