The discussion centers on determining the normalization constant \( c \) for the probability function \( f_X(x) = \frac{c}{x!} \) where \( x = 0, 1, 2, \ldots \). The correct value of \( c \) is established as \( c = \frac{1}{e} \) after recognizing that the sum of the series \( \sum_{x=0}^{\infty} \frac{1}{x!} = e \). Additionally, the conversation addresses the definition of discrete random variables, clarifying that the range of \( X \) is countably infinite, thus confirming that it is indeed a discrete variable despite being unbounded.
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Does countable mean finite or countably infinite? It almost surely means countably infinite. The nonnegative integers are easily seen to be countable:buddingscientist said:however I have one small problem, in my studies I've learned "a random variable X will be defined to be discrete if the range of X is countable" - introduction to theory of statistics (mood). but since the values of X was unbounded in the question (X = 0, 1, 2, ...) i.e: Z+ that is uncountable. ?