I assume you don't want to rebind the event, but call the handler.
You can use trigger() to trigger events:
$('#billing_state_id').trigger('change');
If your handler doesn't rely on the event context and you don't want to trigger other handlers for the event, you could also name the function:
function someFunction() {
//do stuff
}
$(document).ready(function(){
//Load City by State
$('#billing_state_id').live('change', someFunction);
$('#click_me').live('click', function() {
//do something
someFunction();
});
});
Also note that live() is deprecated, on() is the new hotness.