I am trying to make a slightly modified version of checkers for a final project, however I am getting a strange error whenever I run anything related to graphics:  Exception in thread "AWT-EventQueue-1" java.lang.NullPointerException     at Board.drawBoard(TacticalCheckers.java:45)     at TacticalCheckers.paint(TacticalCheckers.java:14)     at sun.awt.RepaintArea.paintComponent(RepaintArea.java:264)     at sun.awt.RepaintArea.paint(RepaintArea.java:240)     at sun.awt.windows.WComponentPeer.handleEvent(WComponentPeer.java:358)     at java.awt.Component.dispatchEventImpl(Component.java:4967)     at java.awt.Container.dispatchEventImpl(Container.java:2294)     at java.awt.Component.dispatchEvent(Component.java:4713)     at java.awt.EventQueue.dispatchEventImpl(EventQueue.java:758)     at java.awt.EventQueue.access$500(EventQueue.java:97)     at java.awt.EventQueue$3.run(EventQueue.java:709)     at java.awt.EventQueue$3.run(EventQueue.java:703)     at java.security.AccessController.doPrivileged(Native Method)     at java.security.ProtectionDomain$JavaSecurityAccessImpl.doIntersectionPrivilege(ProtectionDomain.java:76)     at java.security.ProtectionDomain$JavaSecurityAccessImpl.doIntersectionPrivilege(ProtectionDomain.java:86)     at java.awt.EventQueue$4.run(EventQueue.java:731)     at java.awt.EventQueue$4.run(EventQueue.java:729)     at java.security.AccessController.doPrivileged(Native Method)     at java.security.ProtectionDomain$JavaSecurityAccessImpl.doIntersectionPrivilege(ProtectionDomain.java:76)     at java.awt.EventQueue.dispatchEvent(EventQueue.java:728)     at java.awt.EventDispatchThread.pumpOneEventForFilters(EventDispatchThread.java:201)     at java.awt.EventDispatchThread.pumpEventsForFilter(EventDispatchThread.java:116)     at java.awt.EventDispatchThread.pumpEventsForHierarchy(EventDispatchThread.java:105)     at java.awt.EventDispatchThread.pumpEvents(EventDispatchThread.java:101)     at java.awt.EventDispatchThread.pumpEvents(EventDispatchThread.java:93)     at java.awt.EventDispatchThread.run(EventDispatchThread.java:82)   My code:  // TacticalCheckers.java import java.awt.*; import java.applet.*; import java.util.*;    public class TacticalCheckers extends Applet {      public void paint(Graphics g){         Board.drawBoard(g);     }  }   class Square {      private Color clr;      public void drawSquare(Color clr,Graphics g,int r,int c){         clr = this.clr;         g.setColor(clr);         g.drawRect(r*90,c*90,(r+1)*90,(c+1)*90);     }  }   class Board {     private static Color green = new Color(167,254,138);     private static Color tan = new Color(222,184,135);      private static Square board[][] = new Square[8][8];      public static void drawBoard(Graphics g){         for(int r = 0; r &lt; 8; r++){             for(int c = 0; c&lt; 8; c++){                 if(((r%2 == 0) &amp;&amp; (c%2 == 0)) || ((r%2 == 1) &amp;&amp; (c%2 == 1))){                     board[r][c].drawSquare(green,g,r,c);                 }else{                     board[r][c].drawSquare(tan,g,r,c);                 }              }           }           }     }   I have tried re-installing the IDE (j-grasp) as well and the jre and jdk, but I still get the same error. I have also gotten very similar errors (the only difference being the file/method names) on other graphics programs, which I know used to work. Does anyone know what this error means or more importantly know how to solve it? Thank you for your time! - Fr3d
What are Null Pointer Exceptions (java.lang.NullPointerException) and what causes them?  What methods/tools can be used to determine the cause so that you stop the exception from causing the program to terminate prematurely?
Deriving the joint probability density function from a given marginal density function and conditional density function We have a multivariate normal vector $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\mathbf{X} =  \left[ \begin{matrix} X_1 \\ X_2 \\ \dot\\\\ \dot\\\\ X_n \end{matrix} \right]$,    $\boldsymbol{\mu} =  \left[ \begin{matrix} \mu_1 \\ \mu_2 \\ \dot\\\\ \dot\\\\ \mu_n \end{matrix} \right]$, and  $\boldsymbol{\Sigma} =  \begin{bmatrix} \sigma_1^2 &amp; \sigma_1\sigma_2 &amp; \dots &amp; \sigma_1\sigma_n   \\ \sigma_2\sigma_1 &amp; \sigma_2^2 &amp; \dots &amp; \sigma_2\sigma_n  \\ \dot\\\\ \dot\\\\ \sigma_n\sigma_1 &amp; \sigma_n\sigma_2  &amp; \dots &amp; \sigma_n^2 \end{bmatrix} $ is the covariance matrix.  Y is a random scalar of the form $Y = \beta_0 + \boldsymbol{\beta^TX} + \epsilon$, where $\epsilon \sim \mathcal{N}(0, \sigma_\epsilon^2)$  How can we go about deriving the joint probability density function(pdf) of the random vector $(Y, \boldsymbol{X^T})$?  The marginal multivariate pdf of $\boldsymbol{X}$ is given by  $ pdf(\boldsymbol{x}) = \frac{1}{\sqrt{(2\pi})^n. \begin{vmatrix} \boldsymbol{\Sigma} \end{vmatrix}}.exp\left[ \frac{-1}{2}\boldsymbol{(x-\mu)}^T \boldsymbol{\Sigma^{-1}}\boldsymbol{(x-\mu)} \right]$  The conditional probability density function of $Y \,\mid \, \boldsymbol{X=x}$ is given by  $pdf(y\mid \boldsymbol{X=x}) = \frac{1}{2\pi\sigma_\epsilon}.exp\left[  \frac{-1}{2\sigma_\epsilon^2}(y - \beta_0 - \boldsymbol{\beta^Tx})^2  \right]   $  To compute the joint probability density function, we can use the fact that   $pdf(y, \boldsymbol{x})=  pdf(y\mid \boldsymbol{X=x}).pdf(\boldsymbol{x})$  I am unable to do the witty manipulation required in the algebra to arrive at the result the joint distribution will have a probability density function of a normal distribution. 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