Applied Statistics and Probability for Engineers

example, the response could be the outcome of a functional electrical test on a semiconductor

device for which the results are either a “success,” which means the device works properly, or

a “failure,” which could be due to a short, an open, or some other functional problem.

Suppose that the model has the form

(S11-6)

and the response variable Yitakes on the values either 0 or 1. We will assume that the response

variable Yiis a Bernoulli random variablewith probability distribution as follows:

Yi  0      1 xii

Now since the expected value of the response variable is

This implies that

This means that the expected response given by the response function E(Yi)  0      1 xiis

just the probability that the response variable takes on the value 1.

There are some substantive problems with the regression model in Equation S11-6. First,

note that if the response is binary, the error terms (^) ican only take on two values, namely,
Consequently, the errors in this model cannot possibly be normal. Second, the error variance
is not constant, since
Notice that this last expression is just
since. This indicates that the variance of the observations (which is
the same as the variance of the errors because Yi
i, and (^) iis a constant) is a function
of the mean. Finally, there is a constraint on the response function, because
0 E 1 Yi 2 
i 1
(^) i
E 1 Yi 2  0  1 xi
i
^2 yiE 1 Yi 231 E 1 Yi 24

i 11 
i 2
 11 
i (^22) i 10 
i 2211 
i 2
^2 yiE 5 YiE 1 Yi 262
(^) i 1 0  1 xi 2 when Yi 0
(^) i 1  1 0  1 xi 2 when Yi 1
E 1 Yi 2  0  1 xi
i

i
E 1 Yi 2  (^1 1) i 2  0 11 
i 2
E (^1) i 2 0,
11-7
Yi Probability
1
0 P 1 yi 02  1 
i
P 1 yi 12 
i
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