STOCHATIC MODELS

Please answer the wole question. (show all steps).

Question # II An on-line production of printed-circuit boards (PCBs) involves ten successive sequences (i = 1, 2, 3, ..., 10) of component-placement/assembly on the board. In a quality-control effort, the board is tested for its integrity in each of ten successive assembly-tasks. It is presumed that the reliability of final, completed PCB depends on possible, failure- proneness associated with each (successive) assembly-task involved. Table given below shows the number of PCBs (n.) per successive batch tested. Also shown in the table, is the random number of boards failed in the assembly-line (depicting the set of random variables, RV: x.) in each of the ten successive batches tested. Successively done testing sequence (1 = 1 to 10 batches) $1 #2 # 3 #4 #5 #6 #7 #8 #9 #10 Number of PCBs tested (n.) in 1000 1500 1750 2000 2250 3000 1250 750 2100 1300 each assembly batch Number of failed boards: RV 10 50 25 33 40 38 20 45 152 44 (x.) in each batch tested Determine the following: (i) Most appropriate average performance measure expressed in terms of AM or HM or GM values of the failures involved: Why? (ii) Expected mean: E[.], median, mode, variance and standard deviation () values of the RV (iii) Skewness of the RV expressed in terms of Pearson-1 skewness coefficient; and, specify the type of skewness in terms of relevant parameters that decide the type (left or right-handed) of asymmetry (based on the positions of average measures marked) (iv) Plot (discrete) probability density function (pdf) as a function of RV and construct the associated cdf graph. Marking the mode value (XM.) on the cdf plot, indicate the range of RV from {x, = (XM. -); to {x, =(XMo +)} with respect to the mode and estimate approximate value of the associated cumulative probability in that range of RV (v) Using the cdf-plot, determine the cumulative probability of RV falling in excess of: x. 2 (E[.] + ) and decide on approximate value of this tail-end cumulative probability of the RV (Answer hints: AM ~ 35.7; E[.] ~ 36.9; and, P . ~ - 0.094)