1. An appliance company wants to investigate the life expectancy and reliability of a certain dishwasher. We use a probability density function f(x), where f(x) 0 for all x and where f(x) da represents the probability that the dishwasher breaks down during the time interval from x = a to x = b, where x is measured in years. For the life expectancy of a dishwasher, we use the interval [0, ) because we do not consider negative time. f(x) d dx = (a) (3 marks) A probability density function on the interval [0, ) needs to have the property that 1, so that the total probability is 1 (meaning 100%). Suppose the tech company estimates that the life expectancy of the dishwasher has probability density function fa(x) = ke-0.07x for some value of the constant k. Find the value of k that makes fa(x) function. = ke-0.07x a probability density (b) (3 marks) One measure of 'average' is the mean. The mean (denoted ) of a random variable with probability density function f(x) on [0, ) is defined by = xf(x) dr. Find the mean life expectancy of the dishwasher (using your fa(x) from part (a)). Convert your answer to number of years and months (round to the nearest month). (c) (2 marks) Find the probability of the dishwasher lasting 10 years or longer round to 2 decimal places, or equivalently an integer percent. Optional (not part of the assignment): Compare your answers to parts (b) and (c). Do your answers surprise you? Note that there are several different ways to measure 'average'! (d) (2 marks) The Reliability function R(t) of the dishwasher is defined as the probability that the dishwasher lasts at least t years. Find the Reliability function for your fa(x) from part (a).