Is there a ghost?

I have been having unusual happenings at my house lately. Last night I went to turn on my far bedroom light and it was burnt out, which is not too weird. I had only been using it because my bedside lamp burned out the day prior. So with these two lights out, I then turned on the main light which is an old ceiling fan. The ceiling fan was on too, which is NEVER on. I can't even reach the pull chain to turn it on, and I don't use it because makes a squeaky sound. I use the light every day but never the fan.

Much creepier was tonight. I was on my way upstairs and the front door was open about two feet wide. I didn't even come home through that door and walked by it many times. I thought there must be an intruder. I asked the puppies to check my bedroom for me but I ended up having to do it myself.

A few weeks ago I came come from work and went upstairs and used the bathroom. I then went downstairs and cleaned my jewelry in my fancy ultrasonic cleaner. About 10 minutes later I was near the bottom of the steps and heard water running. Upstairs the shower was going half blast with the curtain open and water going on the floor- something I definitely would have noticed on my earlier trip to the bathroom. I asked everyone and no one had used the bathroom. Brian says that all poltergeist activity can be explained by ultrasound interference, which may account for my creepy shower. But what of the other events? Is there a ghost?