# Different logistic regression models

I've been revising the concepts of logistic regression and suddenly realized that the probability function of the logistic model in the book ISL looks absolutely different from other sources. For ISL logistic function looks this way:

$P(x) = \frac{e^{(\beta_0 + \beta_1x)}}{1 + e^{(\beta_0 + \beta_1x)}}$

(sorry for my lack of knowledge of acceptable formatting). For other sources (especially, courses) logistic function looks another way:

$P(x) = \frac{1}{1 + e^{(\beta_0 + \beta_1x)}}$

(so-called sigmoid function). Moreover, 2 different approaches are used to estimate the coefficients $\beta_0$ and $\beta_1$: for the first equation we're maximizing the likelihood function, while for another we're minimizing the cost function for this model.

Can you please explain me the difference (in simple words) and tell what approach should be used for what case?

• You're missing a minus sign. In that casw if you divide by the exponentials on both sides of the division you get the same thing. Look at Jul 7, 2016 at 22:59
• Oops, i was going to say Wikipedia. en.m.wikipedia.org/wiki/Sigmoid_function Jul 7, 2016 at 22:59
• Oh, thanks a lot, I've really missed that small -1 in the exponent. Jul 7, 2016 at 23:13
• You should really edit that correction into your post! Sep 6, 2017 at 13:19

(Summarizing answer in comments) They are really the same, after you put in that minus sign you lost. Just multiply in numerator and denominator of first form with $\exp(-\beta_0-\beta_1 x)$.