# Normally distributed X implies logistic regression for Y

• Mar 23rd 2011, 03:36 AM
noob mathematician
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 $\beta=(\mu_1-\mu_0/\sigma^2)$ ?
• Mar 23rd 2011, 09:52 AM
theodds
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.