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1 To test a proportion of defective items one draws a sample Xn population with an unknown 0 € (0, 1). (₂) Give the likelihood function of the sample. (22) Find the m.l.e. and MLE of 0. Hint. (2) We can write the density f(x|0) of each r.v. X as 0² (10)¹-1(0.1) (0), ƒ (x|0) = {' {0² (1. 0, x = 0,1 otherwise. Thus, the likelihood function is fn (₂0) = 0° (1 - 0) 1 (0,1) (0) where = L' (0) = n(i=8); where is the sample mean of x₁,..., n. (X₁,..., Xn) from a Bernoulli 0 = = 2₁+ ... +än. (Explain why.) (22) The value of that maximizes the likelihood function is the same value that maximizes the log of fn (n) since the log function is strictly monotone. So, let L(0) : = Infn (xn|0) = oln0 + (n − o)ln(1 — 0). Then, show that

1 To test a proportion of defective items one draws a sample Xn population with an unknown 0 € (0, 1). (₂) Give the likelihood function of the sample. (22) Find the m.l.e. and MLE of 0. Hint. (2) We can write the density f(x|0) of each r.v. X as 0² (10)¹-1(0.1) (0), ƒ (x|0) = {' {0² (1. 0, x = 0,1 otherwise. Thus, the likelihood function is fn (₂0) = 0° (1 - 0) 1 (0,1) (0) where = L' (0) = n(i=8); where is the sample mean of x₁,..., n. (X₁,..., Xn) from a Bernoulli 0 = = 2₁+ ... +än. (Explain why.) (22) The value of that maximizes the likelihood function is the same value that maximizes the log of fn (n) since the log function is strictly monotone. So, let L(0) : = Infn (xn|0) = oln0 + (n − o)ln(1 — 0). Then, show that

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 1YT: YOUR TURN 1 Repeat Example 1 for the probability density function f(t)=83x3 on [1,2]. EXAMPLE 1...
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It is easily seen that L'(0) changes its sign from positive to negative by passing through which of course is a value from [0, 1]. Thus, the empirical sample mean ō is the m.l.e. Ĵ of 0. [Notice that a more elaborate analysis in which we distinguish the cases with ān other values of from (0, 1) yields the same result. ] Thus, X, is the MLE of the sample. 72 = 1 = 0 orann from all
Transcribed Image Text:It is easily seen that L'(0) changes its sign from positive to negative by passing through which of course is a value from [0, 1]. Thus, the empirical sample mean ō is the m.l.e. Ĵ of 0. [Notice that a more elaborate analysis in which we distinguish the cases with ān other values of from (0, 1) yields the same result. ] Thus, X, is the MLE of the sample. 72 = 1 = 0 orann from all
1 To test a proportion of defective items one draws a sample Xn population with an unknown 0 € (0, 1). (₂) Give the likelihood function of the sample. (22) Find the m.l.e. and MLE of 0. Hint. (2) We can write the density f(x|0) of each r.v. X as 0² (10)¹-1(0.1) (0), ƒ (x|0) = {' {0² (1. 0, x = 0,1 otherwise. Thus, the likelihood function is fn (₂0) = 0° (1 - 0) 1 (0,1) (0) where = L' (0) = n(i=8); where is the sample mean of x₁,..., n. (X₁,..., Xn) from a Bernoulli 0 = = 2₁+ ... +än. (Explain why.) (22) The value of that maximizes the likelihood function is the same value that maximizes the log of fn (n) since the log function is strictly monotone. So, let L(0) : = Infn (xn|0) = oln0 + (n − o)ln(1 — 0). Then, show that
Transcribed Image Text:1 To test a proportion of defective items one draws a sample Xn population with an unknown 0 € (0, 1). (₂) Give the likelihood function of the sample. (22) Find the m.l.e. and MLE of 0. Hint. (2) We can write the density f(x|0) of each r.v. X as 0² (10)¹-1(0.1) (0), ƒ (x|0) = {' {0² (1. 0, x = 0,1 otherwise. Thus, the likelihood function is fn (₂0) = 0° (1 - 0) 1 (0,1) (0) where = L' (0) = n(i=8); where is the sample mean of x₁,..., n. (X₁,..., Xn) from a Bernoulli 0 = = 2₁+ ... +än. (Explain why.) (22) The value of that maximizes the likelihood function is the same value that maximizes the log of fn (n) since the log function is strictly monotone. So, let L(0) : = Infn (xn|0) = oln0 + (n − o)ln(1 — 0). Then, show that
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