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3. A random variable X has probability density function f(x) (a-1)2-α for x > 1. (a)...
A random variable X has probability density function f(x)=(a-1)x^(-a),for x>=1. (a) For independent observations x1,...,xn show...
A random variable X has probability density function f(x)=(a-1)x^(-a),for x>=1. (a) For independent observations x1,...,xn show that the log-likelihood is given by, l(a;x1,...,xn)=nlog(a-1)-a (b) Hence derive an expression for the maximum likelihood estimate for ↵. (c) Suppose we observe data such that n = 6 and 6 i=1 log(xi) = 12. Show that the associated maximum likelihood estimate for ↵ is given by aˆ ↵ =1 .5. logri We were unable to transcribe this image
1. Suppose that r,., n are a random sample having probability density function Here the parameter...
1. Suppose that r,., n are a random sample having probability density function Here the parameter θ > 0. (a) Determine the log-likelihood, (0), and a 1-dimensional sufficient statistic. (b) Show that P(X, S b:0) for f(r;0) given in (1) (c) Suppose now that because of a recurring computer glitch in storing the observations, only a +1 for f(r; random subset of the x, are observed. For the rest of the observations, it is only known that z; < 1/2....
1. Suppose that ri,...,In are a random sample having probability density function Here the parameter 0...
1. Suppose that ri,...,In are a random sample having probability density function Here the parameter 0 >0 (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(Xi b;0)-μ+1 for f(x;0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the ai are observed. For the rest of the observations, it is only known that z; < 1/2. Let δί-1 or 0 according to...
2. i) Let B be a random variable with the Binomial (n, p) distribution, so that...
2. i) Let B be a random variable with the Binomial (n, p) distribution, so that Write down the likelihood function L(p) for m independent observations xi,...,Inm 2 marks 6 marks ili) Compute the bias and the mean squared error of the corresponding maximum likeli- from B. Int ii) Show that the maximum likelihood estimate for pis-Σ.ri. mn [7 marks] hood estimator of p. iv) Let X be a continuous random variable with density function for x > 0, and...
1. Suppose that ri,.., n are a random sample having probability density function Here the paran...
1. Suppose that ri,.., n are a random sample having probability density function Here the paran neter θ > 0. (a) Determine the log-likelihood, (0), and a 1-dimensional sufficient statistio (b) Show that PX, b:0) =&+1 for f(z:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the ai are observed. For the rest of the observations, it is only known that Xi < 1/2. Let δί...
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter...
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter θ > 0. (a) Determine the log likelibood, 10), and a 1- dimensional (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(XiS b;0) = +1 for f(x:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the r, are observed. For the rest of the observations,...
2. A random variable X has probability density fun ction f(x) (A-2)1 -A where λ >...
2. A random variable X has probability density fun ction f(x) (A-2)1 -A where λ > 2. (a) Given observations rn from the distribution of X, find the maximum likelihood estimator λ of λ. (As part of your answer you should verify this is 3 a mazimum likelihood estimator.) (b) If n = 20 and Σ1gxi = 5, show that λ = 6. 1) ー1
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample...
. Suppose that x1, . . . , xn are a random sample having probability density...
. Suppose that x1, . . . , xn are a random sample having probability density function f(x; θ) = (θ + 1)x^θ , 0 < x < (1) Here the parameter θ > 0. (a) Determine the log-likelihood, l(θ), and a 1-dimensional sufficient statistic. (b) Show that P(Xi ≤ b; θ) = b θ+1 for f(x; θ) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of...
(9) [12 pts] An exponentially distributed random variable, call it X, has the following probability density...
(9) [12 pts] An exponentially distributed random variable, call it X, has the following probability density functior f(x)-oe ex , x > 0, θ > 0 Note that ElX] and VX]ー1 For the rest of this question, assume that you have a data set (xn1 consisting of a random sample of N observations of X. (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that...
1. Suppose that r,., n are a random sample having probability density function Here the parameter θ > 0. (a) Determine the log-likelihood, (0), and a 1-dimensional sufficient statistic. (b) Show that P(X, S b:0) for f(r;0) given in (1) (c) Suppose now that because of a recurring computer glitch in storing the observations, only a +1 for f(r; random subset of the x, are observed. For the rest of the observations, it is only known that z; < 1/2....
1. Suppose that ri,...,In are a random sample having probability density function Here the parameter 0 >0 (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(Xi b;0)-μ+1 for f(x;0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the ai are observed. For the rest of the observations, it is only known that z; < 1/2. Let δί-1 or 0 according to...
2. i) Let B be a random variable with the Binomial (n, p) distribution, so that Write down the likelihood function L(p) for m independent observations xi,...,Inm 2 marks 6 marks ili) Compute the bias and the mean squared error of the corresponding maximum likeli- from B. Int ii) Show that the maximum likelihood estimate for pis-Σ.ri. mn [7 marks] hood estimator of p. iv) Let X be a continuous random variable with density function for x > 0, and...
1. Suppose that ri,.., n are a random sample having probability density function Here the paran neter θ > 0. (a) Determine the log-likelihood, (0), and a 1-dimensional sufficient statistio (b) Show that PX, b:0) =&+1 for f(z:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the ai are observed. For the rest of the observations, it is only known that Xi < 1/2. Let δί...
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter θ > 0. (a) Determine the log likelibood, 10), and a 1- dimensional (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(XiS b;0) = +1 for f(x:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the r, are observed. For the rest of the observations,...
2. A random variable X has probability density fun ction f(x) (A-2)1 -A where λ > 2. (a) Given observations rn from the distribution of X, find the maximum likelihood estimator λ of λ. (As part of your answer you should verify this is 3 a mazimum likelihood estimator.) (b) If n = 20 and Σ1gxi = 5, show that λ = 6. 1) ー1
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample...
(9) [12 pts] An exponentially distributed random variable, call it X, has the following probability density functior f(x)-oe ex , x > 0, θ > 0 Note that ElX] and VX]ー1 For the rest of this question, assume that you have a data set (xn1 consisting of a random sample of N observations of X. (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that...
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