A member of the Pareto family of distributions (often used in economics to model income distributions)...
A member of the Pareto family of distributions (often used in economics to model income distributions) has a distribution function given by
F(y)={1−(b/y)a, y≥b
0, elsewhere, where a,b>0 are parameters.
1. Find the density function f(y)
Suppose U has the uniform distribution on the interval [0,1] and
find a function G(a,b,u) so that G(a,b,U) has the Pareto
distribution with parameters a and b.
2. G(a,b,u)=
Homework Answers
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A member of the Pareto family of distributions (often used in
economics to model income distributions)...
The Pareto distribution has been used in economics as a model for a density function with...
The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: f(x|x0, θ) = θx0θ x−θ−1, x ≥ x0,θ > 1 Assume that x0 > 0 is given and that X1,...,Xn is an i.i.d. sample. find a sufficient statistic for θ
A Pareto distribution is often used in economics to explain a distribution of wealth. Let a...
A Pareto distribution is often used in economics to explain a
distribution of wealth. Let a random variable X have a Pareto
distribution with parameter θ so that its probability distribution
function is
for
and 0 otherwise. The parameters and
are
known and fixed; is a constant to
be determined.
a) Assuming that
find the expected value and variance of ?
b) Show that for 3 ≥ θ > 2 the Pareto distribution has a
finite mean but infinite variance,...
6.4.4. The Pareto distribution is frequently used a model in study of incomes and has the...
6.4.4. The Pareto distribution is frequently used a model in study of incomes and has the distribution function F(x;0,2)=1-(81/x)02 elsewhere, where 01 0 and 02 > 0 If X\,X2, ...,Xn is a random sample from this distribu- tion, find the maximum likelihood estimators of 01 and 02.
12. Suppose XIX, iid X, P(θ, l), where P(0,1) is the one-parameter Pareto distribution with density...
12. Suppose XIX, iid X, P(θ, l), where P(0,1) is the one-parameter Pareto distribution with density f(x)-0/10+1 for l < x < 00, Assume that θ >2, so that the model θ/(0-1)(8-2)2 (a) obtain the MME θι from the first moment equation and the MIE θ2 (b) Obtain the asymptotic distributions of these two estimators. (c) Show that the ML is asymptotically superior to the MME P(0,1) has finite mean θ/(9 -1 ) and variance
2. A distribution often used to model the lifetime of electronic components is the Rayleigh density...
2. A distribution often used to model the lifetime of electronic components is the Rayleigh density 0 otherwise where θ > 0. (a) If Y has the Rayleigh density, find the distribution for U-Y2 (b) Use the result of part a) to find EY
The Pareto probability distribution has many applications in economics, biology, and physics. Let β> 0 and...
The Pareto probability distribution has many applications in economics, biology, and physics. Let β> 0 and δ> 0 be the population parameters, and let XI, X2, , Xn be a random sample from the distribution with probability density function zero otherwise. Suppose B is known Recall: a method of moments estimator of δ is δ = the maximum likelihood estimator of δ is δ In In X-in β has an Exponential (0--) distribution Suppose S is known Recall Fx(x) =...
Suppose that Y has pdf given by f(y)=b/(y^2); y>=b, b>0. Suppose also that U has a...
Suppose that Y has pdf given by f(y)=b/(y^2); y>=b, b>0. Suppose also that U has a uniform(0,1) distribution. What must the function g(*) be so that Z=g(U) has the same distribution as Y?
Exercise 6.17. Let U and V be independent, U~ Unif(0,1), and V~ Gamma(2.A) which means that...
Exercise 6.17. Let U and V be independent, U~ Unif(0,1), and V~ Gamma(2.A) which means that V has density function fv(1) λ2e-W for v0 and zero elsewhere. Find the joint density function of (X, Y)-. (UV, ( 1-U)V). Identify the joint distribution of (X, Y) In terms of named distributions. This exercise and Example 6.44 are special cases of
Problem 8: Suppose the Ý, , , Y, β are independent and identically distributed random variables in the interval (0,1) with individual densities where β 〉 0. Further suppose that β has marginal densi...
Problem 8: Suppose the Ý, , , Y, β are independent and identically distributed random variables in the interval (0,1) with individual densities where β 〉 0. Further suppose that β has marginal density f(β) 482 exp(-2β). Derive f(B|Y, Y). Identify the distributional family for B and describe its parameters.
Problem 8: Suppose the Ý, , , Y, β are independent and identically distributed random variables in the interval (0,1) with individual densities where β 〉 0. Further suppose that...
MULTIVARIATE DISTRIBUTIONS 3. Suppose that Xi and X2 are independent and each has a uniform distribution...
MULTIVARIATE DISTRIBUTIONS
3. Suppose that Xi and X2 are independent and each has a uniform distribution on (0,1). Define Y: X1 + X2 and Y2 = X1-X2. Find the marginal probability density functions of Y1 and Y2. . Suppose that X has a standard normal distribution, and that the conditional distribution of Y given X is a normal distribution with mean 2X 3 and variance 12. Find E(Y) and Var(Y)
A Pareto distribution is often used in economics to explain a
distribution of wealth. Let a random variable X have a Pareto
distribution with parameter θ so that its probability distribution
function is
for
and 0 otherwise. The parameters and
are
known and fixed; is a constant to
be determined.
a) Assuming that
find the expected value and variance of ?
b) Show that for 3 ≥ θ > 2 the Pareto distribution has a
finite mean but infinite variance,...
6.4.4. The Pareto distribution is frequently used a model in study of incomes and has the distribution function F(x;0,2)=1-(81/x)02 elsewhere, where 01 0 and 02 > 0 If X\,X2, ...,Xn is a random sample from this distribu- tion, find the maximum likelihood estimators of 01 and 02.
12. Suppose XIX, iid X, P(θ, l), where P(0,1) is the one-parameter Pareto distribution with density f(x)-0/10+1 for l < x < 00, Assume that θ >2, so that the model θ/(0-1)(8-2)2 (a) obtain the MME θι from the first moment equation and the MIE θ2 (b) Obtain the asymptotic distributions of these two estimators. (c) Show that the ML is asymptotically superior to the MME P(0,1) has finite mean θ/(9 -1 ) and variance
2. A distribution often used to model the lifetime of electronic components is the Rayleigh density 0 otherwise where θ > 0. (a) If Y has the Rayleigh density, find the distribution for U-Y2 (b) Use the result of part a) to find EY
The Pareto probability distribution has many applications in economics, biology, and physics. Let β> 0 and δ> 0 be the population parameters, and let XI, X2, , Xn be a random sample from the distribution with probability density function zero otherwise. Suppose B is known Recall: a method of moments estimator of δ is δ = the maximum likelihood estimator of δ is δ In In X-in β has an Exponential (0--) distribution Suppose S is known Recall Fx(x) =...
Exercise 6.17. Let U and V be independent, U~ Unif(0,1), and V~ Gamma(2.A) which means that V has density function fv(1) λ2e-W for v0 and zero elsewhere. Find the joint density function of (X, Y)-. (UV, ( 1-U)V). Identify the joint distribution of (X, Y) In terms of named distributions. This exercise and Example 6.44 are special cases of
Problem 8: Suppose the Ý, , , Y, β are independent and identically distributed random variables in the interval (0,1) with individual densities where β 〉 0. Further suppose that β has marginal density f(β) 482 exp(-2β). Derive f(B|Y, Y). Identify the distributional family for B and describe its parameters.
Problem 8: Suppose the Ý, , , Y, β are independent and identically distributed random variables in the interval (0,1) with individual densities where β 〉 0. Further suppose that...
MULTIVARIATE DISTRIBUTIONS
3. Suppose that Xi and X2 are independent and each has a uniform distribution on (0,1). Define Y: X1 + X2 and Y2 = X1-X2. Find the marginal probability density functions of Y1 and Y2. . Suppose that X has a standard normal distribution, and that the conditional distribution of Y given X is a normal distribution with mean 2X 3 and variance 12. Find E(Y) and Var(Y)
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