3. The Pareto distribution is a common probability distribution for describing quantities thatfollow so-called power laws (e.g., maximum one-day rainfall totals in different locations, thesizes of files transmitted via the Internet, etc.). Here we will use calculus again to learn a bitmore about this distribution.a) Suppose you were given the cdf for this distribution:x < Xmin0,F(x)(Xmin`,х > ХminхDifferentiate the cdf with respect to x to obtain the pdf. [You can check your answer bylooking on Wikipedia.]b) Assuming that a > 1, show that the mean/expected value of this distribution is given by:αχminE(X) :a – 1

a)Given C.D.F for pareto distribution is

Answered: 3. The Pareto distribution is a common…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question * The cumulative distribution function F(x) of a random variable X is given in the following table: 1 0.1 3 F(x) 4. 1 a Let a and b are positive constants. If P(2 3) = 0.2. Then a and b equal to: a30.1, b30.7 a=0.5, 63D0.8 a30.1, 63D0.3 None of these Question*
Suppose that the lifetime in years, X, of a type of electrical switch follows the distribution f(x)=12e−x2f(x)=12e−x2for x>0x>0. What type of continuous distribution is this and what is the average lifetime of one of these switches?
Areas under normal distribution curves can be used calculate probabilities of many random variables in the natural and social sciences. The graph of the Standard Normal Curve, with a mean of 0 and a standard deviation of 1 is given below, and its function is given by 1 f(x) = √2T e 1 The integral of e dx would be used to calculate the probability of an event that occurs √2T -1.5 between -1.5 to 1.5 standard deviations from the mean. (-1.5, 0.13) 1 Unfortunately, √2π -1.5 methods to approximate the values. 1.5 -1.5 1.5 (-1, 0.24) Using Simpson's Rule, (-0.5, 0.35) e dx has no proper closed form antiderivative, so we must use numerical 1.5 -1.5 1-0.5 -0-2 Using the Midpoint Rule with three rectangles of equal width, dx 0 Using the Trapezoidal Rule with six trapezoids of equal width, (0, 0.4) 0.5 Round answers to 4 decimal places when appropriate. (0.5, 0.35) (1, 0.24) 1.5 Ibe 1.5 1.5 [ 13 e-t de = \ е 1.5 e-2 е dx (1.5, 0.13)
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
Weights of Lamb after eight weeks after born are normally distributed with mean µ = 14 kg and variance σ 2 = 4. a)Find the probability p that a single lamb has weight less than 12 kg. b) Find the probability that exactly 3 of the next 5 lambs examined will have weights less than 12 kg.
For each random variable defined, describe the set of possible values for the variable, and state whether the variable is discrete or continuous. (a) U = number of times a surfer has to paddle in front of a wave before catching one (b) X = length of a randomly selected angelfish
Question 2: X ~ Geometric(T) distribution with probability function p(x) = P(X = x) = T(1 – 7)"-1 for x = 1, 2, 3, ... Find P(X x).

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