1. 2. The current (in mA) measured in a piece of copper wire is known to follow a uniform distribution over the interval [0,25]. Write down the formula for the probability density function f (:5) of the random variable X representing the current. Calculate the mean and variance of the distribution and find the cumulative distribution function F(:1:). The thickness :1: of a protective coating applied to a conductor designed to work in corrosive conditions follows a uniform distribution over the interval [20, 40] microns. Find the mean. standard deviation and cumulative distribution function of the thickness of the protective coating. Find also the probability that the coating is less than 35 microns thick. 3. Commonly, car cooling systems are controlled by electrically driven fans. Assuming that the lifetime T in hours of a particular make of fan can be modelled by an exponential distribution with A = 0.0003 find the proportion of fans which will give at least 10000 hours service. If the fan is redesigned so that its lifetime may be modelled by an exponential distribution with A = 0.00035. would you expect more fans or fewer to give at least 10000 hours service? 4. The time intervals between successive barges passing a certain point on a busy waterway have an exponential distribution with mean 8 minutes. (a) Find the probability that the time interval between m successive barges is less than 5 minutes. (b) Find a time interval t such that we can be 95% sure that the time interval between two successive barges will be greater than t