close search
Hit Return to see all results

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?


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: 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:

Step 3


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(...

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Our solutions are written by experts, many with advanced degrees, and available 24/7

See Solution
Tagged in



Related Probability Q&A

Find answers to questions asked by student like you

Show more Q&A add

Q: The legislature of Puerto rico consits of a 27 member senate and a 51 member house of representitive...

A: Given:(a)There are 27+51=78 members of the legislature.Now, we have find the number of ways to choos...


Q: 1.4.12 Let Q(x) be the statement “x+ 1 &gt; 2x.” if the domain consists of all integers, what are th...

A: Given statement Q(x)  "x+1 &gt; 2x"Domain consists of all integers.∃xQ(x) means there exists a value...


Q: For the experiment of rolling an ordinary pair of dice, find the probability that the sum will be od...

A: Concept of probability:Probability deals with the likelihood of occurrence of a given event. The pro...


Q: Inclusions are defects in poured metal caused by contaminants. The number of (large) inclusions in c...

A: Introduction to Poisson distribution:Poisson distribution is most suitable, when the events are “rar...


Q: In a sample of 1100 U.S. adults, 212 dine out at a restaurant more than once per week. Two U.S. adul...

A: Given, Total U.S adults = 1100               Adults who dine out at a restaurent more than once per ...


Q: A drawer contains 17 black socks, 17 white socks, 17 blue socks, and 17 brown socks. If the light is...

A: Let’s say that he has picked four socks. If he have already found a pair after two or three or four ...


Q: suppose that a shipment of 120 electronic components contains 4 defective components. to determine w...

A: In this question it has been given thatProbability that a selected electronic component is defective...


Q: Let Xi be iid Exp(λ) random variables.1. Let Mn = max1≤i≤n Xi. Compute the CDF FMn (x).2. Using the ...

A: 1.Introduction:Here, X1, X2,,…, Xn, are iid (independent and identically distributed) random variabl...


Q: Homework: Homework 8.5 - Probability Distribution; Expected Score: 0 of 1 pt 1 of 6 (0 complete) HW ...

A: Given:The total number of balls= 6 (white balls) + 2(blue balls)= 8 balls.Number of balls can be dra...

Sorry about that. What wasn’t helpful?