Week 6 (12-01-26 to 16-01-26)

Prepare

📖 Read the syllabus

📖 Read the support resources

Participate

🖥️ Lecture 4

🖥️ Statistical Table - Statistical Table for probability estimations.

🖥️ Lecture 3 -Continuous Probability Distributions

🖥️ Lecture 2 - Discrete Probability Distributions

🖥️ Lecture 1 - Introduction to Probability

Practice

  1. The current (in mA) measured in a piece of copper wire is known to follow a uniform distribution over the interval [0, 25].

    1. Write the probability density function (pdf) f(x).
    2. Calculate the mean and variance.

  2. A soft-drink machine is regulated so that it discharges an average of 200 milliliters per cup. If the amount of drink is normally distributed with a standard deviation equal to 15 milliliters,

    1. What fraction of the cups will contain more than 224 milliliters?
    2. What is the probability that a cup contains between 191 and 209 milliliters?
    3. How many cups will probably overflow if 230 milliliter cups are used for the next 1000 drinks?
    4. Below what value do we get the smallest 25% of the drinks?

  3. In the context of the Design and Prototype (ID3020) course offered to 3rd semester engineering students at the Faculty of Engineering, University of Jaffna, a laboratory experiment was conducted to evaluate the quality of electronic components selected for student design projects. During this experiment, students collected and analyzed data on the resistance values of 2400 electronic components used in various prototype circuits. These components underwent thorough testing, and the resistance values were observed to follow a normal distribution with a mean of 118 ohms and a standard deviation of 18 ohms.

    1. Considering the random selection of a component from this dataset, what is the probability that its resistance value falls below 150 ohms?
    2. Delving deeper into the dataset, what specific resistance value corresponds to the 88th percentile of these components?
    3. Adhering to the quality standards for prototype functionality, if components with resistance values below 108 ohms are considered unsuitable, how many components would be disqualified based on this criterion?
    4. Seeking to analyze the distribution comprehensively, between which two resistance values, symmetrically positioned around the mean, do we find that approximately 95% of the resistance values lie?

  4. In electrical engineering, power distribution systems often rely on insulated cables buried underground to ensure safe and efficient transmission. The operational reliability of these cables is critical. Suppose the time T in hours until a particular type of underground insulated power cable fails due to insulation breakdown can be modeled by an exponential distribution with λ = 0.0001

    1. Find the proportion of cables that will operate without failure for at least 25,000 hours.
    2. If the insulation material is improved so that the failure rate changes to λ = 0.00007, would you expect a higher or lower proportion of cables to remain functional for at least 25,000 hours?

  5. In civil engineering, the durability of asphalt pavements is a critical factor in road design and maintenance planning. Suppose the time T in years until a newly constructed asphalt pavement section develops its first major crack can be modeled using an exponential distribution with a failure rate λ = 0.2.

    1. What is the probability that the pavement section develops its first major crack within the first 3 years?
    2. The highway authority aims to have at least 90% of pavement sections last more than a certain number of years before cracking. What is the minimum number of years the pavement must last to meet this requirement?

Exercise

  1. A company produces parts for an engine. Parts specifications suggest that 95% of items meet specifications. The parts are shipped to customers in lots of 100.

    1. What is the probability that more than two items in a given lot will be defective?
    2. What is the probability that more than 10 items in a lot will be defective?

  2. The serum cholesterol level X in fourteen-year-old boys has approximately a normal distribution with a mean of 170 and a standard deviation of 30.

    1. Find the probability that the serum cholesterol level of a randomly chosen fourteen-year-old boy exceeds 230.
    2. In a middle school, there are 300 fourteen-year-old boys. Find the probability that at least eight boys have a serum cholesterol level that exceeds 230.