Probability Distributions in R - Assignment - 12

 

Lab Problems: Probability Distributions in R

Problem 1 — Binomial Distribution: Quality Control

A manufacturing company produces electronic components. Based on previous quality-control records, the probability that a randomly selected component is defective is 0.08.

A quality-control engineer randomly selects 15 components from the production line.

Let $X$ represent the number of defective components among the 15 selected components.

Tasks

Using the Binomial distribution:

1. Identify the parameters $n$, $p$, and $q$.
2. Calculate the probability of getting exactly 2 defective components.
3. Calculate the probability of getting at most 2 defective components.
4. Calculate the probability of getting at least 2 defective components.
5. Generate the complete probability distribution for $X=0,1,\ldots,15$.
6. Calculate the mean, variance, and standard deviation.
7. Plot the probability distribution using a suitable graph.
8. Write a brief interpretation of the results.

Expected distribution: Binomial

Useful R functions for verification: dbinom(), pbinom()

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Problem 2 — Normal Distribution: Student Examination Marks

The marks obtained by students in a university examination are approximately normally distributed with a mean of 65 marks and a standard deviation of 10 marks.

Let $X$ represent the mark obtained by a randomly selected student.

Tasks

Using the Normal distribution:

1. Identify the mean $\mu$ and standard deviation $\sigma$.
2. Calculate the z-score of a student who obtains 80 marks.
3. Find the probability that a randomly selected student scores less than 80.
4. Find the probability that a student scores more than 80.
5. Find the probability that a student scores between 55 and 75.
6. Find the probability that a student scores between 60 and 80.
7. Find the mark corresponding to the 90th percentile.
8. Generate 1000 random observations from the given Normal distribution.
9. Plot the Normal probability density curve.
10. Plot a histogram of the generated marks and compare it with the theoretical Normal distribution.
11. Interpret the results.

Expected distribution: Normal

Useful R functions for verification: dnorm(), pnorm(), qnorm(), rnorm()

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Problem 3 — Poisson Distribution: Help-Desk Requests

A university's computer help desk receives an average of 6 service requests per hour. Assume that requests occur independently and that the average arrival rate remains approximately constant.

Let $X$ represent the number of requests received by the help desk in one hour.

Tasks

Using the Poisson distribution:

1. Identify the parameter $\lambda$.
2. Calculate the probability of receiving exactly 4 requests in one hour.
3. Calculate the probability of receiving at most 4 requests.
4. Calculate the probability of receiving at least 4 requests.
5. Calculate the probability of receiving more than 6 requests.
6. Generate the probability distribution for $X=0,1,\ldots,15$.
7. Calculate the mean, variance, and standard deviation.
8. Generate 1000 random observations from the Poisson distribution.
9. Plot the probability distribution.
10. Plot a histogram of the generated observations.
11. Interpret the results.

Expected distribution: Poisson

Useful R functions for verification: dpois(), ppois(), qpois(), rpois()

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Suggested progression for the three experiments

Experiment	Distribution	Main concept students explore
1	Binomial	Fixed number of independent trials and number of successes
2	Normal	Continuous measurements, z-scores and areas under the curve
3	Poisson	Number of events occurring in a fixed interval