6. Given the probability distribution for number of blemishes in a pieces of fabric, calculate the expected value, variance, and standard deviation. (15 points) X 0 1 2 3 4 P(X)0.350.390.190.06 0.01 7. A particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents an independent trial. (a) Over five mornings, what is the probability that the light is green on exactly one day? (b) Over 20 mornings, what is the probability that the light is green on more than four days? 8. The number of flaws in bolts of cloth in textile manufacturing is assumed to have a mean of 0.1 flaw per square meter. (a) What is the probability that there are two flaws in 1 square meter of cloth? (b) What is the probability that there is one flaw in 10 square meters of cloth? (c) What is the probability that there are at least two flaws in 10 square meters of cloth? 9. The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a standard deviation of 0.1 fluid ounce. (a) What is the probability a fill volume is less than 12 fluid ounces? (b) If all cans less than 12.1 or greater than 12.6 ounces are scrapped, what proportion of cans is scrapped? (c) Determine specifications limits (upper and lower limits) that are symmetric about the mean that include 99% of all cans. **Hint: You are looking for two values here (upper and lower limits). 99% Lower Specification Upper Specification Limit Limit