The professor should give you full credit for this problem even if you have answered it wrong because he wrote it wrong.It was actually 4 choices because the question had another part asking for the support of the random variable. But only these two options for this part of the question.
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Like this.
And honestly no I don't think he looked at the problem. It was an online test and I believe the questions were generated and he only selected them. I sent him an email and he said the answer is the fourth one and I'm not sure if he really checked it himself or looked at the answer key to the test.