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Math Help - Normally distributed X implies logistic regression for Y

  1. #1
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    Normally distributed X implies logistic regression for Y

    Suppose the distribution of X for subjects having Y=1 is normal N(\mu_1,\sigma), and suppose the distribution of X for subjects having Y=0 is normal N(\mu_0,\sigma). Then why is P(Y=1|x) a logistic regression curve?

    And why \beta=(\mu_1-\mu_0/\sigma^2) ?
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  2. #2
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    Note that Y is Bernoulli with probability \pi (say) and we are given that X|Y \sim N(\mu_Y, \sigma^2). Just calculate

    f_{Y|X} (1 | x) = \frac{f_{X|Y} (x|1) f_Y (1)}{f_{X|Y} (x|0) f_Y (0) + f_{X|Y} (x|1) f_Y (1)}

    and see what you get. I haven't done it all the way through, but it looks to me like you will get a function whose inverse is the logit.
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