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The number of connections to a wrong phone number is often modeled by a Poisson distribution. To estimate the unknown parameter of that distribution (the mean number of wrong connections) one observes a sample x₁,..., n of wrong connections on n different days. Assum- ing that σ = ₁ + ... + x₂ > 0, (i) give the likelihood function of the sample; (ii) find the m.l.e. and MLE of A. =t. (i) If ƒ (x|λ) = e¯^, then show that the likelihood function is

The number of connections to a wrong phone number is often modeled by a Poisson distribution. To estimate the unknown parameter of that distribution (the mean number of wrong connections) one observes a sample x₁,..., n of wrong connections on n different days. Assum- ing that σ = ₁ + ... + x₂ > 0, (i) give the likelihood function of the sample; (ii) find the m.l.e. and MLE of A. =t. (i) If ƒ (x|λ) = e¯^, then show that the likelihood function is

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 10E
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The number of connections to a wrong phone number is often modeled by a Poisson distribution. To estimate the unknown parameter λ of that distribution (the mean number of wrong connections) one observes a sample x₁,...,n of wrong connections on n different days. Assum- ing that σ = x₁ + ... + än > 0, (i) give the likelihood function of the sample; (¿¿) find the m.l.e. and MLE of X. Hint. (i) If f(x|x) = e¯^, then show that the likelihood function is 72 ƒn(\n \A) = e¯n^X° /[[xj! (0 = x1 + ... + xn) j=1 (ii) A) (22) L(A) = lnfn (XX) = get it. After finding L'(X), show that L'(X) = 0 iff λ = o/n = in. Now, σ/n is a critical point of L(A). But we can rewrite L' as L'(A) = (n − A) showing that An L' is positive for > < In L' is equal zero when λ = = In: L'is negative when λ > Tn. It proves that is a local and hence the global maximum point of L and therefore of fn (XnX). Thus, 7 = is the m.1.e. of λ and  = Xn is the MLE of X.
Transcribed Image Text:The number of connections to a wrong phone number is often modeled by a Poisson distribution. To estimate the unknown parameter λ of that distribution (the mean number of wrong connections) one observes a sample x₁,...,n of wrong connections on n different days. Assum- ing that σ = x₁ + ... + än > 0, (i) give the likelihood function of the sample; (¿¿) find the m.l.e. and MLE of X. Hint. (i) If f(x|x) = e¯^, then show that the likelihood function is 72 ƒn(\n \A) = e¯n^X° /[[xj! (0 = x1 + ... + xn) j=1 (ii) A) (22) L(A) = lnfn (XX) = get it. After finding L'(X), show that L'(X) = 0 iff λ = o/n = in. Now, σ/n is a critical point of L(A). But we can rewrite L' as L'(A) = (n − A) showing that An L' is positive for > < In L' is equal zero when λ = = In: L'is negative when λ > Tn. It proves that is a local and hence the global maximum point of L and therefore of fn (XnX). Thus, 7 = is the m.1.e. of λ and  = Xn is the MLE of X.
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