Usually in Linear Regression we consider $X$ as a explanatory variable whose columns are $X_1 , X_2 .....X_{p}$ are the variables which we use predict are the independent variable $y$ , we measure these values on a continuous scale,When the dependent variable y is dichotomous such as, Male or Female , Pass or Fail , Malignant or Benign.

When we have dependent variable y is a qualitative, we can indicate it by indicator variable such as

$$y = 0\ \ \ if\ female \\ y = 1 \ \ \ if \ male$$

So

$$y_i = \beta_0 + \beta_1x_{i1}+ \beta_2x_{i2}+.....+ \beta_px_{ip} + \epsilon_i \ \ \ \ \ \ i = 1,2,3,........,n$$

or in the matrix form we can write

$$Y = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ . \\ . \\ y_n \\ \end{bmatrix} \ \ X = \begin{bmatrix} 1 & x_{1,1} & x_{1,2} & x_{1,3} & . &. & x_{1,p}\\ 1 & x_{2,1} & x_{2,2} & x_{2,3} & . &. & x_{2,p}\\ . & . & . & . & . & . & x_{3,p} \\ . & . & . & . & . & . & .\\ . & . & . & . & . & . & .\\ 1 & x_{n,1} & x_{n,2} & x_{n,3} & . & . & x_{n,p}\\ \end{bmatrix} \ \ \beta = \begin{bmatrix} \beta{0} \\ \beta{2} \\ \beta{3} \\ . \\ . \\ \beta_p \\ \end{bmatrix} \epsilon = \begin{bmatrix} \epsilon{1} \\ \epsilon{2} \\ \epsilon{3} \\ . \\ . \\ . \\ \epsilon_n \\ \end{bmatrix}$$

that is

$$Y = X\beta + \epsilon$$

Remember first column of independent variable matrix X is $\underline{1}$ , for the constant $\beta_0$

Our dependent variable y , that we have to predict is indicator suppose it takes two values , assume y follows a bernoulli distribution

$$y_i = 1 \ with \ P(y_i = 1 ) = \pi_i \\ y_i = 0 \ with \ P(y_i = 0 ) = 1-\pi_i$$

Assuming $E(\epsilon_i) = 0$,

$$E(y_i) = 1 \cdot \pi_i + 0 \cdot(1 - \pi_i) = \pi_i \\ E(y_i) = X\beta = \pi$$

where

$$\pi = \begin{bmatrix} \pi_{1} & \pi_{2} & \pi_{3}& . & . \pi_{n}\\ \end{bmatrix}^{T}$$

Now we know in Linear Regression $\epsilon$ is supposed to follow normal distribution , whereas here we cannot suppose $\epsilon$ to follow normal distribution, because here it take only two discrete values

so we have $E(y_i) =\pi_{i} = \beta_0 + \beta_1x_{i1}+ \beta_2x_{i2}+.....+ \beta_px_{ip}$ where $E(y_i) \in [0,1]$ that put bound on the expected value of y

In logistic regression we use Standard logistic function , some people call it a Sigmoid function. It can be given by

$$E(y_i) = \pi_i = \frac{1}{1+e^{-(\beta_0 + \beta_1x_{i1}+ \beta_2x_{i2}+.....+ \beta_px_{ip})}} \tag{1}$$

Our main work in logistic regression our main aim is to predict $\pi$ , the bernoulli parameter for $Y$ , and generally we took decision by $\pi_i$ greater than 0.5 or less than 0.5

Link Function

Usually every model have a link function which relates the linear predictor $\eta_i$ to the mean response $\mu_i$. First of all we have to understand what is linear predictor, it is a systematic component where $\eta_i = E(y \vert x_i)$ ,So if $g( . )$ is a link function then

$g(\mu_i ) = \eta_i \ \ or \mu_i =g^{-1}(\eta_i)$

In the Linear regression this link is a identity link , whereas in the logistic regression $\mu_i = E(y_i) =\pi_{i}$ so the relation between $\pi_i$ and $\eta_i = E(y \vert x_i) = \beta_0 + \beta_1x_{i1}+ \beta_2x_{i2}+.....+ \beta_px_{ip}$ is a logistic regression so

$$g(X\beta) = \pi$$

We have similar equation $\eqref{1}$ we can use that to get link function

$$\pi = \frac{exp(X\beta)}{1+exp(X\beta)} \\ X\beta=\eta = ln(\frac{\pi}{1-\pi})$$

where $\frac{\pi}{1-\pi}$ is odds and its log is known as log-odds ,this transformation is logit transformation.

It is very hard to estimate $\beta$ theoretically , so we choose gradient-descent algorithm for calculation of the parameter