Page 1: Examination Instructions and Probabilities

Examination Instructions

• Institution: Tribhuvan University, Institute of Engineering

• Subject: Probability and Statistics (SH 452)

• Candidates must answer in their own words as much as possible.

• If necessary, assume suitable data.

Problem Set

1. Acceleration Due to Gravity Experiment

   • Data Collection:

     • Student A: 9.5, 10.2, 9.8, 11.0, 9.62, 9.5

     • Student B: 9.91, 8.9, 10.1, 9.5, 10.0, 9.4, 9.1

   • Analysis: Determine which student's results are more uniform.

2. Contracts in Bidding

   • Probability of getting contract A: 1/5

   • Probability of getting contract B: 1/3

   • Probability of getting both contracts: 1/8

   • Questions:

     • What is the probability of getting contract A or B?

     • What is the probability of getting neither contract?

     • What is the probability of getting only contract A?

3. Negative Binomial Distribution

   • Define it and outline its properties.

4. Collision Probability

   • Average collisions at an intersection: 2 per week.

   • Questions:

     • Probability of no collisions in a week?

     • Probability of exactly one collision?

     • Probability of exactly two collisions?

5. Normal Distribution

   • Discuss normal distribution and standard normal distribution. Outline characteristics.

6. Continuous Random Variable

   • Given probability density function: f(x) = \begin{cases} 14 & 0 < x < 1 \ 0 & \text{otherwise} \end{cases}

   • Tasks:

     • Find mean and standard deviation of X.

     • Find probability P(c < X < o).

7. Parameter and Sample Statistics

   • From the population of size 5 (22, 23, 24, 25, 26):

     • a) List all samples of size 3 without replacement.

     • b) Find the distribution of sample mean.

     • c) Calculate the standard error of the sample mean.

8. Survey on Stress Levels

   • Of 1750 students, 15% experienced extreme levels of stress:

     • Sample size 160: 15 reported yes.

     • Calculate the probability that more than 10% reported stress.