(USE R-Studio to solve this question) In a manufacturing process where glass products are produced, defects or bubbles occur, occasionally rendering the piece undesirable for marketing. It is known that, the number of defects follow a Poisson process and on average, 7 of these items produced on a given day has one or more bubbles. Let denote the number of days until 32 items with one or more bubbles are produced. a)What is the distribution of and what are its parameters? Is this a valid probability density function? b)Find the probability that exceeds 12 days. Illustrate the desired probability on the pdf plot. Calculate the probability also by using Poisson distribution. Plot also the pmf of the corresponding Poisson random variable c)Find the probability that is between 14 and 20 days. Illustrate the desired probability on the pdf plot. Calculate the probability also by using Poisson distribution.
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!
(USE R-Studio to solve this question)
In a manufacturing process where glass products are produced, defects or bubbles occur, occasionally rendering the piece undesirable for marketing. It is known that, the number of defects follow a Poisson process and on average, 7 of these items produced on a given day has one or more bubbles. Let denote the number of days until 32 items with one or more bubbles are produced.
a)What is the distribution of and what are its parameters? Is this a valid probability density
b)Find the probability that exceeds 12 days. Illustrate the desired probability on the
c)Find the probability that is between 14 and 20 days. Illustrate the desired probability on the pdf plot. Calculate the probability also by using Poisson distribution.
d)Calculate the mean, variance and standard deviation of .
e)Find the probability that exceeds 12 days using normal approximation. Illustrate the desired probability on the normal pdf plot.
f)Find the probability that is between 14 and 20 days using normal approximation. Illustrate the desired probability on the normal pdf plot.
g)Compare the results obtained without using approximation and the results obtained by using normal approximation. Is this problem suitable for applying normal approximation?
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