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A random variable is called a power law random variable ifP(X > t) = max{1/(t^α), 1}for some α. What is the density of X? Compute the mean of X when α > 1. What happens if α < 1?

Question

A random variable is called a power law random variable if
P(X > t) = max{1/(t^α), 1}
for some α. What is the density of X? Compute the mean of X when α > 1. What happens if α < 1?

check_circleAnswer
Step 1

Power law distribution:

A random variable, X is said to have a power law probability distribution, when its tail function at a particular value x is of the form:

P( X > x ) ~ L(x) ∕ x α,

Where α > 1 and L(x) is any function of x satisfying the condition: limx→∞ L(rx) ∕ L(x) = 1.

Step 2

The CDF:

In this case, the tail function is P( X > t ) = max {t α , 1}.

This expression is a probability measure, indicating that the maximum value must always be 1, that is, the value of P( X > t ) cannot exceed 1.

Thus, max {t α , 1} in this case must always be 1.

The cumulative distribution function (CDF) of X is:

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Step 3

PDF:

Here, P( X > t ) = 1 and P( X < t ) = 0.

As a result, the random variable X is a degenerate, with the entire probability concentrated at X = t, that is, P(...

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