1 Question 7 and 14 From Written Portion:

Poisson PMF, from 0 to 10

plot(dpois(0:10,4.5),type="h",main="Pois PMF,n=10,average is 4.5",
     xlab="Defects",ylab="Probability")

Explanation: The above shows the lollipop or bar chart that is derived from the above binomial PMF distribution. The graph above shows a more so left-sided skew, meaning there is a much higher chance of those number of defects occurring in the production process.

ppois(3,4.5)
## [1] 0.342296

Explanation: We are simply looking for the amount of breakdowns of the robotic arm in a day, therefore, we are finding amount of defects rather than finding the time duration between breakdowns which is more so with poisson distribution. We simply use the above code which finds 3 or fewer defects with an average of defects being 4.5

Poisson PMF, the probability of the time between consecutive breakdowns being less than 2 hours in duration, assuming an 8 hour workday. 4.5 breakdowns averaging. So we do 4.5 divided by 8 hours to find the hourly average. 4.5/8 = .5625

pexp(2, rate = 4.5 / 8)
## [1] 0.6753475

2 Question 11 From Written Portion:

2.0.1 Context: The context is as follows: Automatic bagging machines are designed to seal packages at 200 grams, with a standard deviation of 30 grams. Assume the weights are normally distributed.

  • Plot the normal PDF from 120 to 280, with appropriate title and axes labels and a brief explanation of the plot.  

  • Find the probability that a package is sealed with less than 150 grams, with a brief description of how to find this probability in general. (note: you do not need to use the standard normal)

Normal Distribution PDF Plot:

curve(dnorm(x,200,30),from=120,to=280,main="Normal PDF:mean=200,sd=30",
      xlab="Weight",ylab="Chance")

Explanation: The above normal distribution plot shows a bell-curve representing the probability of the weight distribution in grams across packages in the automatic bagging machine pipeline. Basically, it is the highest likelihood in percentage that the package will weight 200, which is the mean weight in grams, compared to either side of the bell curve extremes which shows tapering.

pnorm(150,200,30)
## [1] 0.04779035

Explanation: Shows the probability of a sealed package being less than 150 grams, it can be found using z score within normal distribution manually, but through code, it is done via the above line, which is simply the range denoting at most or fewer, mean, and the sd value, 30 in this case.