Baby Name Lucy

Do you think it's normal to name a baby Lucy or does it have to be Lucille and call her Lucy for short?  I love the name Lucy but I wonder if it's like naming your son Mike on the birth certificate instead of Michael. Then again some people are named just John and some are named Jonathan....idk
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Is it one of those things where people will constantly be asking her if her name is short for something? Or maybe people will just call her Lucille anyways even though that's not her name?
Thoughts?