Let X denote a random variable for which E[(X-a)^2] exists. Give an example of a distribution of

step 1 So in this question we need to give an example of a discrete random variable x which is degenera

Let X denote a random variable for which E[(X-a)^2] exists....

Let $X$ denote a random variable for which $E\left[(X-a)^{2}\right]$ exists. Give an example of a distribution of a discrete type such that this expectation is zero. Such a distribution is called a degenerate distribution.

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Key Concepts

Degenerate Distribution

A degenerate distribution is one in which the random variable takes a single fixed value with probability one. This means that there is no variability, and all moments (except perhaps higher order moments in trivial cases) reduce to the fixed value. In the context of the expectation E[(X-a)^2] being zero, it implies that X must equal the constant a almost surely.

Expected Value

The expected value, or mean, of a random variable is a measure of its central tendency, calculated as the weighted average of all possible values that the variable can take. It is fundamental in probability theory as it provides a single summary measure of the distribution’s location. In this context, the expected value is used to assess how the random variable deviates from a constant.

Variance

Variance measures the spread or variability of a random variable around its expected value, and is defined as the expected value of the squared deviation from the mean. When the variance is zero, it indicates that the random variable does not deviate at all from its mean, making it a degenerate case. E[(X-a)^2] being zero is essentially a statement that the variance is zero when the fixed value a is taken as the mean.

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