I need help with these problems...

Answer as directed. Show your complete solutions. 1. Suppose that X is a continuous random variable whose probability density function is given by f(x) = C(4x - 2x]) if0 > 1). [4 points] 2. Find the following probabilities if the joint probability density function of X and Y is given by otherwise a. P(X > 1,Y R Find the variance of X. [5 points] 4. A process for producing batches of chalk is such that the bulk density X of a batch of chalk is a normally distributed random variable with mean 0.8g/cc and standard deviation 0.003g/cc. Find (a) P(1X - 0.8| $ 0.006) and (b] c such that P(X s c) = 0.95. [10 points] 5. A machine produces items in a certain plant using a mass-production process. Its record is that 5% of the items produced are defective. If a sample of 1000 items is chosen at random, what is the probability that between 40 and 60, inclusive, of the items in the sample are defective? [5 points] 6. Let X be the time until a computer server connects to your machine and Y be the time until the server authorizes you as a valid user. Each of these random variables measures the wait from a common starting time where X