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
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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