Markov chain converges to Normal Distribution. How to increase standard deviation

Question

I am creating an MC (using the following recursive function). It is about a game that each side has different probs of scoring 1 point, 2 points, 3 points or no score (scoreDifference is the score side A - score side B, time is 100 time units, and timeRemaining is what's left.

f[t, d] =


If[t > 100, 0,
       If[d == scoreDifference && t == timeRemaining, 1,
        p1h*f[t + 1, d - 1] + p2h*f[t + 1, d - 2] + p3h*f[t + 1, d - 3] +
         p1a*f[t + 1, d + 1] + p2a*f[t + 1, d + 2] + p3a*f[t + 1, d + 3] +
         pns*f[t + 1, d]
        ]
       ];

At timeremaining = 0 I get all the possible differences. This converges to a normal distribution (x-axis is the difference, the y-axis is probability). Example:

p1h = 0.0375 # side A
p2h = 0.05625
p3h = 0.0125
p1a = 0.0327083 # side B
p2a=  0.0490625
p3a = 0.0109028
pns = 0.801076 # noone scores

How can I manipulate this in order to achieve greater (or smaller) standard deviation? In this example:

E[x] = 11.5
V[x] = 335.6774
SD[x] = 18.3215

Answer 1

That is clearly obtained by increasing the probabilities for larger differences ($p_3$'s) wrt those for smaller.