Example 5.5. Recall Example 5.2 (section 5.1.2). Now, suppose that the Xis will not be directly observable. Instead, we will only be able to observe whether X>0 or X = 0. For example, drivers may not want to reveal to an insurance company that they've had a number of accidents in the past for this will surely increase their premium, but, to mitigate the risk of the insurance industry, government reg- ulations may require that they have to reveal something. So a compromise is reached to balance privacy and transparency-drivers must reveal whether they've had any accident at all (i.e. whether X; > 0 or X; = 0), although they need not reveal exactly how many they've had. Perhaps somewhat surprisingly, we can still estimate the parameter 0 in this case, despite the added complication! The key insight here is that we can define another (fully observable) random variable, 1, X; > 0, Y₁ = 0, X = 0. ~ Then, Y; Binomial(1, pi), where = Pi P(Y - 1) = P(X; > 0) = 1 - P(X = 0) = e -Ovi (Ovi)º = 1. = 1 − e−vi 0! Table 5.1 Number of individuals (out of 40) who had zero (X; = 0) or at least one (X; > 0) incident, along with their level of activity (v;) One+ Zero (X; > 0) (X = 0) Vi 8 10 0 4 2 1 237 1873 Source: authors.

Example 5.5. Recall Example 5.2 (section 5.1.2). Now, suppose that the Xis will not be directly observable. Instead, we will only be able to observe whether X>0 or X = 0. For example, drivers may not want to reveal to an insurance company that they've had a number of accidents in the past for this will surely increase their premium, but, to mitigate the risk of the insurance industry, government reg- ulations may require that they have to reveal something. So a compromise is reached to balance privacy and transparency-drivers must reveal whether they've had any accident at all (i.e. whether X; > 0 or X; = 0), although they need not reveal exactly how many they've had. Perhaps somewhat surprisingly, we can still estimate the parameter 0 in this case, despite the added complication! The key insight here is that we can define another (fully observable) random variable, 1, X; > 0, Y₁ = 0, X = 0. ~ Then, Y; Binomial(1, pi), where = Pi P(Y - 1) = P(X; > 0) = 1 - P(X = 0) = e -Ovi (Ovi)º = 1. = 1 − e−vi 0! Table 5.1 Number of individuals (out of 40) who had zero (X; = 0) or at least one (X; > 0) incident, along with their level of activity (v;) One+ Zero (X; > 0) (X = 0) Vi 8 10 0 4 2 1 237 1873 Source: authors.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 64E

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