please solve it

NOTE: Attempt all questions. Q.#1(a) A bolt is manufactured by 3 machines A, B and C. A turns twice as many items as b and machine B and C produce equal number of items. Two percent of bolts produced by A and B are defective and 4% of bolts produced by C are defective. All bolts are put into one stock pile and one is chosen from this pole. What is the probability that it is defective? (b)A manufacturer of a flu vaccine is concerned about the quality of its flu serum. Batches of serum are processed by three different departments having rejection rates of 0.10, 0.08, and 0.12 respectively. The inspection by the departments are sequential and independent. (i) What is the probability that a batch of serum survives the second department inspection but is rejected by the third department? (ii)What is the probability that a batch of serum is rejected by all the department? Q.#2 (a) If the probability mass function of a random variable is given by P(X = r) = kr,r = 1,2,3, f, find (i)the value of k. (i) PC >). (iii) the mean and variance of X. (b) The CDF F(x) of a random variable X is given by: 0, if xs1 F(x) = c(x - 1)*, if 1 3 (i) find PDF f(x) (ii) P(IX| 0. find a such that P([X| 1). Also find P(|X - 1| 2). (b)If X has a uniform distribution in (0, 3) ant Y has exponential distribution with parameter a, find a that Var (X) = Var(Y). Also find P(Y > 0.5/Y