The Central Limit Theorem states: Let X1,... , X,, be a random sample with E[X,] = H and Var(X) = o?. If n is sufficiently large, then X has approximately a normal distribution with mean uy = µ and variance o? = o?/n. It's also true that X, ~ N(nu, nơ2). Suppose a machine requires a specific type of battery that lasts an exponential amount of time with mean 25 hours. As soon as the battery fails, you replace it immediately. If you have 50 such batteries, estimate the probability that the machine is still operating after 1300 hours. Round your answer to three decimal places. Save your answer as p4.
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!
![Problem 4
The Central Limit Theorem states: Let X1, ..., Xn be a random sample with E[X;] = µ and Var(X¡) = o?. If n is sufficiently large, then X has
approximately a normal distribution with mean ui = µ and variance o? = o?In. It's also true that E X, ~ N(nu, no²).
Suppose a machine requires a specific type of battery that lasts an exponential amount of time with mean 25 hours. As soon as the battery fails, you replace it
immediately. If you have 50 such batteries, estimate the probability that the machine is still operating after 1300 hours. Round your answer to three decimal
places. Save your answer as p4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b86bad7-f2a1-4808-85a5-0e00eff97186%2F62575de1-e80c-4cb4-a703-0512eab0a872%2Fuu93je_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
