When creating an instance variable for a method, why doesn't Jason adds it to the initialize method?

Not sure if this answers your question but you are on the right track. First attr_reader automatically creates a first_and_middle_name method that just returns the value of first_and_middle_name. Ruby does this behind the seems and it'll look something like:

def first_and_middle_name
  @first_and_middle_name
end

The question is where does that instance variable come from. Well, you are right that it could (and maybe should) be set in the initializer, since you have all the data there to build it. You could do something like:

def initialize(title, first_name, middle_name, last_name)
  @title = title
  @first_name = first_name
  @middle_name = middle_name
  @last_name = last_name
  @first_and_middle_name = @first_name + " " + @middle_name
end

But instead, Jason builds it in the full_name function (notice that the last line of the above initialize method is the same as the line from the video).

So before you call the first_and_middle_name function, @first_and_middle_name doesn't exist! You can try it out by modifying the code like:

name = Name.new("Mr.", "Jason", "", "Seifer")
# puts name.full_name_with_title
puts name.first_and_middle_name.inspect # => nil

If you run that, it'll return nil because full_name was never called to initialize the value. In short, it's somewhat of an oversight made in order to get the main point of the video across.