Week 5 (05-01-26 to 09-01-26)

Prepare

📖 Read the syllabus

📖 Read the support resources

Participate

🖥️ Lecture 3 -Continuous Probability Distributionss

🖥️ Statistical Table - Statistical Table for probability estimations.

🖥️ Lecture 2 - Discrete Probability Distributions

🖥️ Lecture 1 - Introduction to Probability

Practice

  1. In the MC3020 course at the Faculty of Engineering, University of Jaffna, a serious concern was raised regarding impersonation through forged signatures on attendance sheets. To ensure academic integrity, an investigation was launched focusing on the authenticity of student signatures recorded during lectures. A total of 240 student signatures were randomly sampled and analyzed using advanced signature verification techniques. The analysis revealed that the similarity scores between the submitted signatures and verified authentic samples followed a normal distribution with a mean of 112 and a standard deviation of 18. A higher similarity score indicates a greater likelihood that the signature is genuine.

    1. What is the probability that a randomly selected signature has a similarity score below 152, potentially indicating authenticity?
    2. What similarity score corresponds to the 95th percentile, representing signatures highly likely to be genuine?
    3. If signatures with similarity scores below 105 are flagged as potentially forged, how many of the 240 sampled signatures are expected to be flagged?
    4. Between which two similarity scores, symmetrically positioned around the mean, do approximately 90% of the signature similarity scores fall, representing the normal range for genuine signatures?

  2. Dr T. Mayooran is organizing a critical midterm review session for the MC3020 Probability and Statistics lectures, which has ten students. Each student has a 60% chance of attending the session independently. The Lecturer needs to calculate several probabilities to: - Decide whether to book a small classroom (max 4 seats) or a larger one. - Prepare enough group activity materials (requires at least 2 students). - Gauge the likelihood of high attendance for distributing resources.

    1. What is the probability that exactly six students attends the session?
    2. What is the probability that at most three students attend (i.e., the small classroom is sufficient)
    3. The session requires at least two students to run group activities. What is the probability the session proceeds as planned?

  3. A radar-based meteor detection system is used at the Computer Engineering Research Lab to monitor meteor activity. The system is configured to detect meteors appearing randomly and independently in the sky. Historical data shows that, on average, 1.50 meteors are detected in any 30-second interval under specified conditions.

    1. What is the probability that no meteors are detected in a one-minute interval?
    2. What is the probability of observing at least five but not more than eight meteors in two minutes of observation?
    3. Suppose the system is modified such that each meteor detection has a 10% chance of being a false positive (i.e., not an actual meteor). If 10 meteors are detected in a 5-minute interval, what is the probability that exactly 2 of them are false positives?
    4. In a dataset of 100 meteor detections, 15 are known to be rare events. If 10 detections are randomly selected for analysis, what is the probability that at least 3 of them are rare events?

  4. A discrete random variable X taking values 0,1,2. . . . . . ,6 has probability mass function given by

    X = x 0 1 2 3 4 5 6
    P(X = x) 3k 2k k 0 2k 4k 6k

    for some constant k.

    1. Find the value of k.
    2. Calculate P(X > 1).
    3. Calculate E(X).
    4. Calculate the variance of X.
    5. Find the variance of the random variable Y = 9 − 2X.

Exercise

  1. In the November 1990 issue of Chemical Engineering Progress, a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was 99.61 with a standard deviation of 0.08. Assume that the distribution of percent purity was approximately normal.

    1. What percentage of the purity values would you expect to be between 99.5 and 99.7?

    2. What purity value would you expect to exceed exactly 5% of the population?

  2. High accumulation of certain pollutants in the atmosphere can adversely affect the environment and human health, which is a concern for the Ariviyal Nagar region, particularly the vicinity of the University of Jaffna. Elevated levels of pollutants such as particulate matter, sulphur dioxide, and nitrogen oxides can lead to respiratory issues and environmental degradation. Nitrogen oxides (NOx) are primary contributors to air pollution, primarily originating from vehicle emissions, industrial processes, and the combustion of fossil fuels. Suppose the distribution of nitrogen oxide emissions from vehicles in the Ariviyal Nagar region can be modelled adequately by a normal distribution with a mean emission level (μ) of 80 ppb (parts per billion) and standard deviation(σ) of 12 ppb.

    1. What is the probability that a randomly selected vehicle from the Ariviyal Nagar region will have emission levels less than 50 ppb?

    2. What is the probability that a randomly selected vehicle from the Ariviyal Nagar region will have emission levels greater than 100 ppb?

    3. What is the probability that a randomly selected vehicle from the Ariviyal Nagar region will have emission levels between 40 ppb and 116 ppb?

  3. The operator of a pumping station has observed that demand for water during early afternoon hours follows an approximately exponential distribution with a mean of 100 cubic feet per second (cfs).

    1. Find the probability that the demand will exceed 200 cfs during the early afternoon on a randomly selected day.

    2. What water-pumping capacity should the station maintain during the early afternoons so that the probability that demand will exceed capacity on a randomly selected day is only 0.01?

  4. The heights of 1000 students are approximately normally distributed with a mean of 174.5 centimetres and a standard deviation of 6.9 centimetres. Suppose 200 random samples of size 25 are drawn from this population and the means are recorded to the nearest tenth of a centimetre. Determine

    1. the mean and standard deviation of the sampling distribution of \(\bar{X}\).

    2. the number of sample means falling below 172.0 centimeters.

    3. the number of sample means that fall between 172.5 and 175.8 centimeters inclusive.

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