Please answer each of the following questions in detail and provide examples for better clarity wherever applicable. Provide in-text citations.

What makes a function of a discrete variable a candidate for a discrete random variable distribution? What about the counterpart of this candidacy in the case of a continuous variable?

Provide a detailed discussion on the distribution of:A discrete random variable, in general terms, and then provide a numerical example of this distribution.

What are the mean and the standard deviation in your example?

How does this differ in the case where the random variable is continuous?

Explain the significance of the mean, variance, and standard deviation for a random variable.

How does the probability of a union of disjoint events exhibit itself when dealing with a (discrete or continuous) random variable? Provide an example

What is the expectation operation and what are its properties? How does the expectation operation yields relate between the mean, the standard deviation, and the second moment?

PLEASE DO NOT PLAGIARISE

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