# Quantization step-size for a given signal based on its PDF

I am familiar with the principles of midtread and midrise quantizer. However, I have difficulties determining the step size where it hasn't been explicitly given. For example, the following probability distribution of the input signal s(t) is given. This signal shall be quantized with 2 bit and we want to use a uniform, midrise quantizer. How can I calculate the step size and thus reconstruction values/decision thresholds for this quantizer?

You have a uniform input PDF, and the optimal quantizer for a uniform inpu will be a uniform quantizer. Note that the your uniform PDF has two pieces and the interval from $$-4$$ to $$4$$ is completely untreated.
Then, assuming a 2-bit (4 level) uniform midrise quantizer with the given PDF your decision intervals $$I_k$$ and reconstruction levels $$y_k$$ will be as follows:
$$x \in I_1 = [-6,-5] \implies y_1 = -5.5$$ $$x \in I_2 = [-5,-4] \implies y_2 = -4.5$$ $$x \in I_3 = [4,5] \implies y_3 = 4.5$$ $$x \in I_4 = [5,6] \implies y_4 = 5.5$$