(a) Given that X is normally distributed with mean 10 and ?(? > 12) = 0.1587. What is the probability that X will fall in the interval (9, 11)? You are given: Φ(1) = 0.8413 and Φ(−1/2)=0.3085, where Φ(?) = ?(−∞ < ? < ?)such that ?~?(0,1). (b) Suppose that during rainy season on a tropical island the length of the shower has an exponential distribution, with parameter ? = 2, time being measured in minutes. What is the probability that a shower will last more than three minutes? If a shower has already lasted for 2 minutes, what is the probability that it will last for at least one more minute?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
(a) Given that X is
(b) Suppose that during rainy season on a tropical island the length of the shower has an exponential distribution, with parameter ? = 2, time being measured in minutes. What is the probability that a shower will last more than three minutes? If a shower has already lasted for 2 minutes, what is the probability that it will last for at least one more minute?
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