How to get more space in piecewise function on LaTeX?

Don't use $$ for display math.

\displaystyle is not needed.

Option -1:

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\[
P(Y_i = j) = \left\{
        \begin{array}{ll}
            \frac{1}{1+\sum_{k=1}^{K-1}e^{\beta_k\cdot x_i}} & \quad k=K \\[1em]
            \frac{e^{\beta_j\cdot x_i}}{1+\sum_{k=1}^{K-1}e^{\beta_k\cdot x_i}} & \quad 1 \leq k \leq K-1
        \end{array}
    \right.,
\]
\end{document}

Option - 2

You can use cases.

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\[
P(Y_i = j) = \begin{cases}
            \frac{1}{1+\sum_{k=1}^{K-1}e^{\beta_k\cdot x_i}} & \quad k=K \\[1em]   %%% <--- here
            \frac{e^{\beta_j\cdot x_i}}{1+\sum_{k=1}^{K-1}e^{\beta_k\cdot x_i}} & \quad 1 \leq k \leq K-1
        \end{cases}
\]
\end{document}

In either case, you can use \\[<dimen>] to change the vertical separation.

Beautification

To make your sums to look less cramped, you can use \limits as in

\sum\limits_{k=1}^{K-1}

Code:

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\[
P(Y_i = j) = \begin{cases}
            \frac{1}{1+\sum\limits_{k=1}^{K-1}e^{\beta_k\cdot x_i}} & \quad k=K \\[2em]
            \frac{e^{\beta_j\cdot x_i}}{1+\sum\limits_{k=1}^{K-1}e^{\beta_k\cdot x_i}} & \quad 1 \leq k \leq K-1
        \end{cases}
\]
\end{document}

I had no idea that amsmath had such a useful redefinition, so that should be the correct answer (after use of cases). However, for completeness (or if you want more space) you have a few options:

• reformulate your terms to allow for more vertical leg-room
• modify \arraystretch to a desired scale (but will cause potentially awkward space on top)

• use a strut, as in the example below:

  \documentclass{article}
  \usepackage{array}
  
  %\renewcommand\arraystretch{4}
  
  \begin{document}
  \[
  P(Y_i = j) = \left\{
    \begin{array}{>{\displaystyle}ll}
      \biggl(1+\sum_{k=1}^{K-1}e^{\beta_k\cdot x_i}\biggr)^{-1}
      & \quad k=K \\
      \biggl(1+\sum_{k=1}^{K-1}e^{\beta_k\cdot x_i}\biggr)^{-1}
      e^{\beta_j\cdot x_i}
      & \quad 1 \leq k \leq K-1
  
      \rule{0pt}{8ex} % called a 'strut' -- a vertical rule of nil width
    \end{array}
    \right.,
  \]
  \end{document}
  

  

The dcases environment by »mathtools« from the »mh« bundle seems to be quite handy here.

\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{mathtools}  % loads »amsmath«

\begin{document}
  \[
    P(Y_i=j)=
    \begin{dcases}
      \frac{1}{1+\sum\limits_{k=1}^{K-1}e^{\beta_k\cdot x_i}} & \quad k=K \\
      \frac{e^{\beta_j\cdot x_i}}{1+\sum\limits_{k=1}^{K-1}e^{\beta_k\cdot x_i}} & \quad 1 \leq k \leq K-1
    \end{dcases}
  \]
\end{document}

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