A company produces apple juice bottles with a volume of approximately 3 liters each. One machine fills half of each bottle with concentrate, and another machine fills the other half with water. Assume the two machines work independently. The volume (in liters) of concentrate poured by the first machine follows a normal distribution with mean 0.96 and variance 0.0007. The volume of water (in liters) poured by the second machine follows a normal distribution with mean 1.02 and variance 0.0018. A bottle of orange juice produced by this company is therefore a mixture of water and concentrate. What is the probability that a bottle contains more than 2.12 liters of juice?
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 company produces apple juice bottles with a volume of approximately 3 liters each. One machine fills half of each bottle with concentrate, and another machine fills the other half with water. Assume the two machines work independently. The volume (in liters) of concentrate poured by the first machine follows a
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