Problem 1 Assume the following Cumulative probability function of X, the number of defects. F(x)= 0 X=3 Using only F(x), calculate : P(X=1.5) Problem 2 A batch of 500 machined parts contains 10 that do not conform to customer requirements. The random variable is the number of parts in a sample of three parts that do not conform to customer requirements. -Give the possible values of X -Give the probability mass function of X -VVhat is the probability that more than 1 out of 3 do not conform to requirements. Problem 3 1.25 1.5 1.75 2 2.25 (x) 0.2 0.4 0.1 0.2 0.1 -Is this a valid probability mass function? -Calculate the cumulative probability (F(x) function of the days until change. -P(X>2) -P(X=3 Problem 5 Let X be a binomial random variable with p = 0.1 and n = 10. Calculate the following probabilities. 1. P(X s 2) 2. P(X > 8) 3. P(X = 4) . P(5 1. Determine the following: 1. P(X 5) 3. P(4 8) 5. x such that P(X 1.5) 3. P(X 6) 5. Variance Problem 11 In 2002, the average height of a woman aged 20-74 years was 64 inches, with an increase of approximately 1 inch from 1960. Suppose the height of a woman is normally distributed with a standard deviation of 2 inches. 1. What is the probability that a randomly selected woman in this population is between 58 inches and 70 inches? 2. What are the quartiles (25th and 75th percentiles) of this distribution? 3. Determine the height that includes the lowest 90% of this population. 4. Determine the height that is symmetric about the mean that includes 90% of this population. 5. What is the probability that five women selected at random from this population all exceed 68 inches? 6. What is the probability that if five women selected at random from this population, their Average height will exceed 68 inches?Problem 12 An article in Mathematical Biosciences [\"Inuence of Delayed Viral Production on Viral Dynamics in HIV-1 Infected Patients\" (1998, Vol. 152(2), pp. 143163)] considered the time delay between the initial infection by immunodeficiency virus type 1 (HIV-1) and the formation of productively infected cells. In the simulation model, the time delay is approximated by a gamma distribution with parameters r = 4 and A = 4 days. Determine the following: 1. Mean and variance of time delay 2. Probability that a time delay is more than half a day 3. Probability that a time delay is between one-half and one day 4. What is the probability that the average delay in 40 infections is less than 2 days