QUESTION TWO (20 marks) a) The probability that a light bulb lasts for less than x hours is 1-e. Obtain the moment generating [4 Marks] function of x. b) A company produces two lots of products, that is; products which are highly liked by customers (lot A) and products which are fairly liked by customers (lot B). if 20% of the products produced by the company are highly liked by the customers, find the probability that, 10 products selected at random there are, i.) At least two are highly liked. ii.) Find the mean and the variance. i.) ii.) iii.) c) An important factor in solid missile fuel is the particle size distribution. Significant problems occur if the particle size are too large. From production in the past, it has been determined that the particle size (in micrometers) distribution is characterized by [3x, x>1 f(x)=0, elsewhere Verify that this is a valid density function Evaluate F(x) [2 marks] [2 marks] d) A random variable X follows a normal distribution with a mean of 8 and a variance of 49 i.) Find P(|X-8