Randomly sample 137 pair (x, y) from y = ax^3 + bx^2 +cx+d (choose a,b,c,d by yourself). Use the sampled y as mean values to build 137 normal distributions (choose the variances randomly by yourself). Then sample one y value each from the 137 random distributions. (50 points)

a. Use the 137 pairs (x, y, s) to perform a weighted least square fit with y = a3x^3 + a2 x^2 +a1 x + a0 (the estimate of the error for each a is also needed). Calculate chi square and analyze the results.

b. Make a plot to show the data (x, y, s) and the line y = a3x^3 + a2 x^2 +a1 x + a0 .