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?

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: lim_{x}_{→∞}* 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:

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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