Distributions HW

It would be very helpful if you could plot the distributions before calculating the probabilities. Begin with reading up on the plot() function.

These questions will help you build an understanding of Normal, Binomial, Hypergeometric and Poisson distribution. Some online readings may be helpful as well.

You will be using the probability density function, cumulative density function and quantile function in this assignment.

Q2. A quality control inspector has drawn a sample of 13 light bulbs from a recent production lot. Suppose 20% of the bulbs in the lot are defective. What is the probability that less than 6 but more than 3 bulbs from the sample are defective?

Round your answer to four decimal places.

i. Identify the distribution. This is a binomial distribution problem. The binomial distribution is used to describe the number of successes in a fixed number of trials, and here the success is defined as bulb being defective. You can think of the bulb being “special” and not necessarily bad/“failure” from a manufacturing perspective if that helps.

• Check its assumptions quickly.
  • There are only two possible outcomes on a particular trial of an experiment (bulb defective or not).
  • The outcomes are mutually exclusive.
  • The random variable is the result of counts.
  • Each trial is independent of any other trial…

ii. Identify the parameters

\(n=13\)

\(\pi = 20\% =.2\)

iii. See what is the Probability Statement Required to be Solved so that you know what to compute

Let X be the count of defective bulbs.

In Words : Probability that less than 6 but more than 3 bulbs from the sample are defective.

In Math: \(P(3<X<6 | n=13, \pi =.2 )\)

iv. Compute/Evaluate

plot(x    = 0:13, 
     y    = dbinom(x    = 0:13, 
                   size = 13, 
                   prob = .2
                  ), 
     type = 'h',
     main = 'Binomial Distribution (n=13, p=0.2)',
     ylab = 'Probability',
     xlab = '# Successes',
     lwd  = 3
     )

dbinom(x = 4:5, size = 13, prob = .2)

## [1] 0.15354508 0.06909529

sum(dbinom(x = 4:5, size = 13, prob = .2))

## [1] 0.2226404

round(x = sum(dbinom(x = 4:5, size = 13, prob = .2)),digits = 4)

## [1] 0.2226

0.2226