Probability & Statistics for Engineers & Scientists 9th edition 9.82 Consider the lognormal distribution with the...
Probability & Statistics for Engineers & Scientists 9th edition
9.82
Consider the lognormal distribution with the density function given in Section 6.9. Suppose we have a random sample x1,x2,....,xn from a lognormal distribution.
(a) Write out the likelihood function.
(b)Develop the maximum likelihood estimates of mean and variance of random variable.
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Probability & Statistics for Engineers & Scientists 9th
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9.82
Consider the lognormal distribution with the...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random var...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
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) =...
QUESTION 5 Suppose that Yı, Y2,.., Yn independent variables such that where β is an unknown...
QUESTION 5 Suppose that Yı, Y2,.., Yn independent variables such that where β is an unknown parameter, X1, x2-.., xn are known real numbers, and el,e2 independent random errors each with a normal distribution with mean 0 and variance ơ2 ,en are (a) Show that is an unbiased estimator of β. What is the variance of the estimator? (b) Given that the probability density function of Y is elsewhere, show that the maximum likelihood estimator of β is not the...
Let X1, X2,.. .Xn be a random sample of size n from a distribution with probability...
Let X1, X2,.. .Xn be a random sample of size n from a distribution with probability density function obtain the maximum likelihood estimator of θ, θ. Use this maximum likelihood estimator to obtain an estimate of P[X > 4 when 0.50, 2 1.50, x 4.00, 4 3.00.
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability fun...
Advanced Statistics, I need help with (c) and (d)
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x;...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3e-dız?, x > 0. a. Find E(X), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for \, Gamma for the function, and pi for the mathematical constant 11. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/ I. Hint 1: Consider u = 1x2 or u = x2....
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x;...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3 e-tz, x > 0. a. Find E(XK), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for 1, Gamma for the function, and pi for the mathematical constant i. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/n. Hint 1: Consider u = 1x2 or u = x2....
Suppose that X1, X2, ..., Xn is an iid sample, each with probability p of being...
Suppose that X1, X2, ..., Xn is an iid sample, each with probability p of being distributed as uniform over (-1/2,1/2) and with probability 1 - p of being distributed as uniform over (a) Find the cumulative distribution function (cdf) and the probability density function (pdf) of X1 (b) Find the maximum likelihood estimator (MLE) of p. c) Find another estimator of p using the method of moments (MOM)
Let X1, X2,... X,n be a random sample of size n from a distribution with probability...
Let X1, X2,... X,n be a random sample of size n from a distribution with probability density function obtain the maximum likelihood estimator of λ, λ. Calculate an estimate using this maximum likelihood estimator when 1 0.10, r2 0.20, 0.30, x 0.70.
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
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) =...
QUESTION 5 Suppose that Yı, Y2,.., Yn independent variables such that where β is an unknown parameter, X1, x2-.., xn are known real numbers, and el,e2 independent random errors each with a normal distribution with mean 0 and variance ơ2 ,en are (a) Show that is an unbiased estimator of β. What is the variance of the estimator? (b) Given that the probability density function of Y is elsewhere, show that the maximum likelihood estimator of β is not the...
Let X1, X2,.. .Xn be a random sample of size n from a distribution with probability density function obtain the maximum likelihood estimator of θ, θ. Use this maximum likelihood estimator to obtain an estimate of P[X > 4 when 0.50, 2 1.50, x 4.00, 4 3.00.
Advanced Statistics, I need help with (c) and (d)
2. Let X1, X2, ..., Xn be a random sample from a Bernoulli(6) distribution with prob- ability function Note that, for a random variable X with a Bernoulli(8) distribution, E [X] var [X] = θ(1-0) θ and (a) Obtain the log-likelihood function, L(0), and hence show that the maximum likelihood estimator of θ is 7l i= I (b) Show that dE (0) (c) Calculate the expected information T(e) EI()] (d) Show...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3e-dız?, x > 0. a. Find E(X), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for \, Gamma for the function, and pi for the mathematical constant 11. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/ I. Hint 1: Consider u = 1x2 or u = x2....
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3 e-tz, x > 0. a. Find E(XK), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for 1, Gamma for the function, and pi for the mathematical constant i. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/n. Hint 1: Consider u = 1x2 or u = x2....
Suppose that X1, X2, ..., Xn is an iid sample, each with probability p of being distributed as uniform over (-1/2,1/2) and with probability 1 - p of being distributed as uniform over (a) Find the cumulative distribution function (cdf) and the probability density function (pdf) of X1 (b) Find the maximum likelihood estimator (MLE) of p. c) Find another estimator of p using the method of moments (MOM)
Let X1, X2,... X,n be a random sample of size n from a distribution with probability density function obtain the maximum likelihood estimator of λ, λ. Calculate an estimate using this maximum likelihood estimator when 1 0.10, r2 0.20, 0.30, x 0.70.
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