The cumulative distribution function for a certain random variable X is given as follows: F (x) = {0, x < 0 4x^3 - 3x^4, 0 lessthanorequalto x lessthanorequalto 1 1, x greaterthanorequalto 1. (a) Find P(x greaterthanorequalto 1/2). (b) Find the pdf, f (x). (F (x) = integral^x _-infinity f (t)dt.) (c) Find the mean and variance. 20 percent (20%) of tree light bulbs manufactured by a company are defective. The company's Quality Control Manager is quite concerned and therefore randomly samples 50 bulbs coming off of the assembly line. Let X denote the number in the sample that are defective. What is the probability that the sample contains at most four defective bulbs? The results of our MIDTERM in MATH 279 are as follows: 75, 8, 36, 36, 55, 55, 27, 83, 17, 58, 55, 42, 36, 50, 42, 82, 27, 92, 50, 42, 100, 83, 27, 58, 55 Find a 95% confidence interval for the average of all tests at NJIT. The probability density function (pdf) for the length of a part in millimeters is f (x) = 30e^-30 (x - 10) for x > 10 in and zero for x lessthanorequalto 10 in. Although the intended target length is 10 in several factors can produce lengths exceeding 10 in. (a) Determine the mean and variance of the length. (b) The probability that length exceeds 10.1 in. (c) Find the cumulative function, F(x).