solve the following questions: 1) The number of telephone calls arriving at an exchange during any given minute between noon and 1:00pm is a random variable X with the following probability distribution f(x):

x	0	1	2	3	4	5
f(x)=P(X=x)	0.3	0.2	0.2	0.1	0.1	0.1

a) Verify that f(x) is a probaility distribution. b) Find the cumulative distribution function F of the random variable X at x = 0, 1, 2, 3, 4 and 5. c) Use F(x) to find the probality that more than two calls will arrive within a minute. d) Find the expected value (or mean), the variance and the standard deviation of X. 2) The probability that the noise level of a wide -band amplifyer will exceed 2dB is 0.05. Find the probability that among 12 such amplifyers a) one will exceed 2dB; b) at most two will exceed 2db; c) two or more will exceed 2dB. d) Find the mean and variance of the number of amplifyers exceeding 2dB in the sample. 3) An airline company knows by experience, that passengers reserving a seat have a 5% probability of not showing up at the gate. Therefore the company sells 257 tickets for a plane with only 255 seats. What is the probability that every passenger arriving gets a seat? Hint: You can speed up calculations by using the formula in exercise 3 above. 4) A shipment of 120 burglar alarms contains 5 that are defective. If 3 of those are randomly selected and shipped, calculate the probability that there is one defective alarm using: a) the formula for the hypergeometric distribution; b) the binomial distribution as an approximation. a) What are the expected number and variance of defective alarms in the shipment?