You have all you need to know in the comments to your question.
Assuming you are using the Gregorian calendar, then in its cycle of $400$ years there are $97$ leap years and so $365\times 400+97 = 146097$ days, which is exactly $20871$ weeks, so each cycle repeats the weekdays. All you need to do is count $400$ consecutive Christmases; you could count $2800$ or some other multiple, but it would not change the proportions.
Since $400$ is not divisible by $7$, there is no possibility that each weekday will appear the same number of times. In fact you get the following numbers:
Divide each of these by $400$ and you get the proportions in your question.
There is no particular reason why Sunday, Tuesday and Friday are most common; they just are. Some day(s) had to be more common than others since $400$ is not divisible by $7$; in the previously used Julian calendar, each weekday appeared four times for 25 December every $28$ years.