title: How can I program a random walk in Mathematica?
body: <p>In a plane (two dimensional), a path can be represented by a sequence of (n+1) points (Xo,Yo),(X1,Y1),...,(Xn,Yn) such that, for any i (integer 1 &lt; i &lt; n-1):</p>
  
  <pre><code>Pi(vector) = [Xi-X(i-1),Yi-Y(i-1)]
  </code></pre>
  
  <p>representing the ith step, is a vector with the length Pi, and a value of the change of direction between the vectors Pi and P(i+1) is measured algebraically (don't know how) by the turning angle a(i).</p>
  
  <p>Like any angular distribution (of changes of direction) it is characterized by a mean vector which is taken to be symmetrical and to have angular mean Φ = o. The approach to this analysis involves numerical simulations, since the algebraic approach seems to be too complex and I have to use a pseudo-random Gaussian generator to obtain continuous values from a normal distribution with a mean of 0 and a standard deviation σ (0.1-1.2)radians to simulate a path.</p>
  
  <p>So after each step with the length P (is constant i.e 125km), the value of the change of direction (turning angle a(i)) is determined by the pseudo-random generator for a given value of σ, which is constant along the path. And then makes a step in the next direction, and so on.</p>
  
  <p>Some useful equations:</p>
  
  <pre><code>a(i) ~ n(0,σ)
  Θ(i+1) = Θ(i) + a(i)
  X(i+1) = Xi + P Cos[Θ(i+1)]
  Y(i+1) = Yi + P Sin[Θ(i+1)]
  </code></pre>
  
  <p>where Θi represents the direction of the ith step. The direction of the first step Θi, is chosen at random according to a uniform angular distribution by a pseudo-random uniform generator. Turning angles are recorded from -Pi to Pi</p>
  
  <h3>So my question is:</h3>
  
  <p>How can I take i.e 12 families of 500 step paths each characterized by a given value of standard variation σ ranging between 0.1 and 1.2 radians, of the distribution of changes of direction between successive steps and PLOT it in Mathematica? I don't know anything about Mathematica specially how  to write the code for this problem.</p>