Week 6 (26-05-25 to 30-05-25)

Prepare

📖 Read the syllabus

📖 Read the support resources

Participate

🖥️Lecture 3 - Continuous Probability Distributions

🖥️ Statistical Table - Statistical Table for probability estimations.

🖥️ Lecture 2 - Discrete Probability Distributions

🖥️ Lecture 1 - Introduction to Probability

Practice

  1. A discrete random variable X taking values 0,1,2,3,4,5,6 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

  2. A large company has an inspection system for the batches of small compressors purchased from vendors. A batch typically contains 15 compressors. In the inspection system, a random sample of 5 is selected and all are tested. Suppose there are 3 faulty compressors in the batch of 15.

    1. What is the probability that for a given sample there will be 1 faulty compressor?
    2. What is the probability that the inspection will reveal more than one faulty compressor in the sample?

  3. The current (in mA) measured in a piece of copper wire follows a uniform distribution over the interval [0, 25].

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

  4. The time intervals (in minutes) between successive barges passing a certain point on a busy waterway are modeled by an exponential distribution with mean of 8 minutes.

    1. Find is the probability that the time interval between two successive barges is less than 5 minutes?
    2. Find a time t such that we can be 95% sure that the time interval between two successive barges will be greater than t.

  5. A lawyer commutes daily from his suburban home to his midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed.

    1. What is the probability that a trip will take at least 1/2 hour?

    2. If the office opens at 9:00 A.M. and the lawyer leaves his house at 8:45 A.M. daily, what percentage of the time is he late for work?

    3. If he leaves the house at 8:35 A.M. and coffee is served at the office from 8:50 A.M. until 9:00 A.M., what is the probability that he misses coffee?

    4. Find the length of time above which we find the slowest 15% of the trips.

    5. Find the probability that 2 of the next 3 trips will take at least 1/2 hour.