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What is the expected value of a constant variable?

Answer 1

The constant itself.

If #X# is a random variable which can only assume the value #x#, then you have

#\mathbb{E}(X)=\mu = x#

This makes sense, since #X# assumes only one value, and so "on average" it assumes that value as well. Think for example that #X# equals constantly #3#. Then you sample, for example, ten values from #X#, and you will have #x_1 = 3#, #x_2 = 3, ... , x_10=3#, since you can't have anything but #3#. Now, compute the average:

#\mu = \frac{x_1 + ... + x_10}{10} = \frac{3+...+3}{10} = \frac{10\cdot 3}{10} = 3#

To be more precise, you may use the definition

#\mathbb{E}(X) = \sum p_i x_i#

i.e. the weighted sum of all possible values, weighted with their probabilities. Since #X# only assumes the value #x# with probability #1#, you have

#\mathbb{E}(X) = \sum p_i x_i = 1\cdot x = x#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.