A probability distribution functioSuppose you are conducting a quality control test on a man- ufacturing process that produces bolts. The length of these bolts follows a normal distribution with an unknown mean and a known standard deviation of 2 millime- ters. We want to decide if = 3 (decision d0) or = 4 (decision d1) based on given 25 samples. If we estimate = 3 when actual = 4, the repair cost of each unit is $4. On the other hand, if we estimate = 4 when actual = 3, the repair cost of each unit is $8. Find the optimal decision rule for using the binary search.n of a discrete random variable X is parameterized by p and r, and it models the number of trials in a sequence of i.i.d. distributed Bernoulli trials with parameter p until r number of successes occurs. You can take the following as facts: P(x)=x1pr(1p)xr r1 E[X] = r(1p) p V ar[X] = r(1p) p2 x1, . . . , xn are n i.i.d observations of this random variable. (a) Using method of moments, estimate p and r. (7 points) (b) Say