Why is it always reasonable to use the normal distribution to solve probability questions regarding a sample mean based on a sample from any arbitrary distribution when the sample size is large?

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Why is it always reasonable to use the normal distribution to solve probability questions regarding a sample mean based on a sample from any arbitrary distribution when the sample size is large?

1. Why is it always reasonable to use the normal distribution to solve probability questions regarding a sample mean based on a sample from any arbitrary distribution when the sample size is large? Which famous theorem justifies the use of the normal distribution in the above situation? 2. When is an estimator called an unbiased estimator? 3. What do we mean by an unbiased estimator? Why is this important?

 

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