For example, a fixed scalar is equal to a gaussian random variable which has its mean and zero variance?
Hi Huiping,
A normal variable with zero variance, I don't think it's very pretty but I understand what you mean. I think it is more appropriate to deal with distribution functions when you have "jumps". But to answer clearly your question, it can. Note for example that, if X_n --> constant c (in distribution) is equivalent to X_n -- > constant c (in probability) ?
As you can see, in this theorem, constant c is clearly considered as a random variable. Generally, one uses the term of degenerate random variable to decribe a random variable that can take only one value.
Best regards !
If the deterministic quantity has value a, then it can be regarded as a (degenerate) random variable whose probability distribution is a point mass of weight 1 at a. Its distribution function is F(x) = 0 for xa. Its density function is delta(x-a).
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