a. Explanation Given info: The data shows the number of hops before the first flight and proceeded by a flight. The total observed frequency is 130 and the experimenter had fitted a geometric distribution for the data. Calculation: Geometric distribution: P ( X = x ) = p x − 1 ⋅ q Where, q is ( 1 − p ) Maximum likelihood estimator: π i ( p ) = f ( x 1 , x 2 , ... , x n ) = ( p x 1 − 1 ⋅ q ) ( p x 2 − 1 ⋅ q ) , ... , ( p x n − 1 ⋅ q ) = p ∑ x i − n ( 1 − p ) n Differentiating, d d p ( ∑ x i − n ) ln ( p ) + n ln ( 1 − p ) = ∑ x i − n p − n ( 1 − p ) Equate the above to 0 yields, ∑ x i − n p − n ( 1 − p ) = 0 ∑ x i − n p = n ( 1 − p ) ∑ x i − n = n p 1 − p ∑ x i − p ∑ x i − n + n p = n p b. To determine Find the expected frequency using p ^ from part (a). Test the fit of the model by combining the observed counts for 7, 8, 9,…12.

Probability and Statistics for Engineering and the Sciences

9th Edition

ISBN: 9781305251809

Author: Jay L. Devore

Publisher: Cengage Learning

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Question

Chapter 14.2, Problem 14E

To determine

Show that the maximum likelihood estimate of p is p^=(∑xi−n)∑xi.

Find the value of p^ for the given data.

To determine

Find the expected frequency using p^ from part (a).

Test the fit of the model by combining the observed counts for 7, 8, 9,…12.

Expert Solution & Answer

Chapter 14 Solutions

Probability and Statistics for Engineering and the Sciences

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Show that the maximum likelihood estimate of p is p ^ = ( ∑ x i − n ) ∑ x i . Find the value of p ^ for the given data.

Solution Summary: The author explains how the experimenter fitted a geometric distribution for the data.

Show that the maximum likelihood estimate of p is p^=(∑xi−n)∑xi.

Find the value of p^ for the given data.

Find the expected frequency using p^ from part (a).

Test the fit of the model by combining the observed counts for 7, 8, 9,…12.

Chapter 14 Solutions

Show that the maximum likelihood estimate of p is p ^ = ( ∑ x i − n ) ∑ x i . Find the value of p ^ for the given data.

Solution Summary: The author explains how the experimenter fitted a geometric distribution for the data.

Show that the maximum likelihood estimate of p is p^=(∑xi−n)∑xi. Find the value of p^ for the given data.

Find the expected frequency using p^ from part (a). Test the fit of the model by combining the observed counts for 7, 8, 9,…12.

Chapter 14 Solutions

Show that the maximum likelihood estimate of p is p^=(∑xi−n)∑xi.

Find the value of p^ for the given data.

Find the expected frequency using p^ from part (a).

Test the fit of the model by combining the observed counts for 7, 8, 9,…12.