solve the following questions: 1) Determine for a continuous random variable X with probability density f(x) = { x for 0 < x < 1 , 2-x for 1 x < 2, 0 otherwise a) P(0.62 (variance). Hint: The easiest way to solve c) is using the earlier discussed identity V(X) = E(X²) - E(X)² 2) If a random variable has a normal distribution, what is the probability that it will take on a value within 2 standard deviations from the mean? 3) The time to microwave a bag of popcorn may be treated as normally distributed with a standard deviation of 10 seconds. If the probability is 0.8212 that the bag will take less than 282.5 seconds to pop, find the probability that it will take longer than 258.3 seconds to pop. 4) Find the quartiles -z_0.25, z_0.50, and z_0.25of the standard normal distribution. 5) Assuming that 30% of all industrial accidents in a certain plant are caused by failure of employees to follow instructions, find, approximately, the probability that among 84 industrial accidents in this plant anywhere from 20 to 30 (inclusive) will be due to failure of employees to follow instructions. 6) A consulting engineer receives, on average, 0.7 requests per week. If the number of requests follows a Poisson process, find the probability that a) in a given week, there will be at least 1 request; b) in a given 4-week period there will be at least 3 requests. 7) Consider waiting for an event for a time t in a Poisson process with on average aarrivals per time unit. a) By considering P(N(t)=0), argue that P(waiting time is > t) = exp(-at). b) Using your result in a), determine the probability density of the waiting time between successive arrivals in a Poisson process with on average a arrivals per time unit.