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(8) Logistic Regression. In logistic regression, we have a binary dependent variable Y (that takes on the values 0 and 1) and a quantitative independent variable X. Let ?(x) = P(Y = 1|X = x). We consider the model Y = E(Y|X = x) + ?. (a) (10 Points) Show that E(Y|X = x) = ?(x), E(?) = 0 and Var(?) = ?(x)(1 - ?(x)) and if log(?(x) / (1 - ?(x))) = ? + ?x, (1) then ?(x) = exp(? + ?x)/(1 + exp(? + ?x)). (b) (10 Points) Suppose the data (x1, y1), (x2, y2), ..., (xn, yn) are observed. Using equation (1) and the likelihood function L((x1, y1), ..., (xn, yn)) = ?(x1)^y1(1 - ?(x1))^(1-y1) ? ... ? ?(xn)^yn(1 - ?(xn))^(1-yn), show that the maximum likelihood estimators for ? and ? can be found by solving the equations ?_{i=1}^n (yi - ?(xi)) = 0 and ?_{i=1}^n xi(yi - ?(xi)) = 0. (2) (c) (10 Points) Given the data (1.0, 0), (2.0, 1), (2.0, 0), (3.0, 1), (3.0, 1), (3.0, 0), (4.0, 1), (4.0, 1), (5.0, 1), verify that the MLE are ?? = -4.15183 and ?? = 1.7831, and predict the probability of Y = 1 if x = 6.0. Hint: Solving the equations in (2) requires use of Mathematica or Wolfram Alpha.

          (8) Logistic Regression. In logistic regression, we have a binary dependent variable Y (that takes on the values 0 and 1) and a quantitative independent variable X. Let ?(x) = P(Y = 1|X = x). We consider the model
Y = E(Y|X = x) + ?.
(a) (10 Points) Show that E(Y|X = x) = ?(x), E(?) = 0 and Var(?) = ?(x)(1 - ?(x)) and if
log(?(x) / (1 - ?(x))) = ? + ?x, (1)
then ?(x) = exp(? + ?x)/(1 + exp(? + ?x)).
(b) (10 Points) Suppose the data (x1, y1), (x2, y2), ..., (xn, yn) are observed. Using equation (1) and the likelihood function
L((x1, y1), ..., (xn, yn)) = ?(x1)^y1(1 - ?(x1))^(1-y1) ? ... ? ?(xn)^yn(1 - ?(xn))^(1-yn),
show that the maximum likelihood estimators for ? and ? can be found by solving the equations
?_{i=1}^n (yi - ?(xi)) = 0 and ?_{i=1}^n xi(yi - ?(xi)) = 0. (2)
(c) (10 Points) Given the data
(1.0, 0), (2.0, 1), (2.0, 0), (3.0, 1), (3.0, 1), (3.0, 0), (4.0, 1), (4.0, 1), (5.0, 1),
verify that the MLE are ?? = -4.15183 and ?? = 1.7831, and predict the probability of Y = 1 if x = 6.0.
Hint: Solving the equations in (2) requires use of Mathematica or Wolfram Alpha.

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(8) Logistic Regression. In logistic regression, we have a binary dependent variable Y (that takes on the values 0 and 1) and a quantitative independent variable X. Let ?(x) = P(Y = 1|X = x). We consider the model
Y = E(Y|X = x) + ?.
(a) (10 Points) Show that E(Y|X = x) = ?(x), E(?) = 0 and Var(?) = ?(x)(1 - ?(x)) and if
log(?(x) / (1 - ?(x))) = ? + ?x, (1)
then ?(x) = exp(? + ?x)/(1 + exp(? + ?x)).
(b) (10 Points) Suppose the data (x1, y1), (x2, y2), ..., (xn, yn) are observed. Using equation (1) and the likelihood function
L((x1, y1), ..., (xn, yn)) = ?(x1)^y1(1 - ?(x1))^(1-y1) ? ... ? ?(xn)^yn(1 - ?(xn))^(1-yn),
show that the maximum likelihood estimators for ? and ? can be found by solving the equations
?i=1^n (yi - ?(xi)) = 0 and ?i=1^n xi(yi - ?(xi)) = 0. (2)
(c) (10 Points) Given the data
(1.0, 0), (2.0, 1), (2.0, 0), (3.0, 1), (3.0, 1), (3.0, 0), (4.0, 1), (4.0, 1), (5.0, 1),
verify that the MLE are ?? = -4.15183 and ?? = 1.7831, and predict the probability of Y = 1 if x = 6.0.
Hint: Solving the equations in (2) requires use of Mathematica or Wolfram Alpha.

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Allan G. Bluman 9th Edition

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01/13/2023

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Logistic Regression. In logistic regression, we have a binary dependent variable Y (that takes on the values 0 and 1) and a quantitative independent variable X. Let ̴̵̶̷̸̡̢̧̨̛̖̗̘̙̜̝̞̟̠̣̤̥̦̩̪̫̬̭̮̯̰̱̲̳̹̺̻̼͇͈͉͍͎̀́̂̃̄̅̆̇̈̉̊̋̌̍̎̏̐̑̒̓̔̽̾̿̀́͂̓̈́͆͊͋͌̕̚ͅ͏͓͔͕͖͙͚͐͑͒͗͛ͣͤͥͦͧͨͩͪͫͬͭͮͯ͘͜͟͢͝͞͠͡ͰͱͲͳʹ͵Ͷͷ͸͹ͺͻͼͽ;Ϳ΀΁΂΃΄΅Ά·ΈΉΊ΋Ό΍ΎΏΐΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡ΢ΣΤΥΦΧΨΩΪΫάέήίΰαβγδεζηθικλμνξοπ(x) = P(Y = 1|X = x). We consider the model Y = E(Y|X = x) + ̴̵̶̷̸̡̢̧̨̛̖̗̘̙̜̝̞̟̠̣̤̥̦̩̪̫̬̭̮̯̰̱̲̳̹̺̻̼͇͈͉͍͎̐̑̒̓̔̽̾̿̀́͂̓̈́͆͊͋͌̕̚ͅ͏͓͔͕͖͙͚͐͑͒͗͛ͣͤͥͦͧͨͩͪͫͬͭͮͯ͘͜͟͢͝͞͠͡ͰͱͲͳʹ͵Ͷͷ͸͹ͺͻͼͽ;Ϳ΀΁΂΃΄΅Ά·ΈΉΊ΋Ό΍ΎΏΐΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡ΢ΣΤΥΦΧΨΩΪΫάέήίΰαβγδεζηθικλμνξοπρςστυφχψωϊϋόύώϏϐϑϒϓϔϕϖϗϘϙϚϛϜϝϞϟϠϡϢϣϤϥϦϧϨϩϪϫϬϭϮϯϰϱϲϳϴϵ϶ϷϸϹϺϻϼϽϾϿε. (a) Show that E(Y|X = x) = π(x), E(ε) = 0 and Var(ε) = π(x)(1 - π(x)) and if log(π(x)/(1 - π(x))) = α + βx, then π(x) = exp(α + βx)/(1 + exp(α + βx)). (b) Suppose the data (x1, y1), (x2, y2), ..., (xn, yn) are observed. Using equation (1) and the likelihood function L((x1, y1), ..., (xn, yn)) = π(x1)^y1(1 - π(x1))^(1-y1) ⋅ ... ⋅ π(xn)^yn(1 - π(xn))^(1-yn), show that the maximum likelihood estimators for α and β can be found by solving the equations ∑(yi - π(xi)) = 0 and ∑xi(yi - π(xi)) = 0. (c) Given the data (1.0, 0), (2.0, 1), (2.0, 0), (3.0, 1), (3.0, 1), (3.0, 0), (4.0, 1), (4.0, 1), (5.0, 1), verify that the MLE are α̂ = -4.15183 and β̂ = 1.7831, and predict the probability of Y = 1 if x = 6.0. Hint: Solving the equations in (2) requires use of Mathematica or Wolfram Alpha.

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