2.6 Exercises
Exercise 2.1 Complete the crossword puzzle found here. Turn in your completed puzzle.
Exercise 2.2 Consider the following discrete distribution of the random variable \(X\) whose probability mass function is \(p(x)\).
| \(x\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(p(x)\) | 0.3 | 0.2 | 0.2 | 0.1 | 0.2 |
Write a KSL program to generate 4 random variates from this distribution using stream 1.
Exercise 2.3 Suppose that customers arrive at an ATM via a Poisson process with mean 7 per hour. Write a KSL program that outputs the arrival time of the first 6 customers using stream 1.
Exercise 2.4 The service times for a automated storage and retrieval system has a shifted exponential distribution. It is known that it takes a minimum of 15 seconds for any retrieval. The rate parameter of the exponential distribution is \(\lambda = 45\). Write a KSL program to generate 10 observations from this distribution using stream 1.
Exercise 2.5 The time to failure for a computer printer fan has a Weibull distribution with shape parameter \(\alpha = 2\) and scale parameter \(\beta = 3\). Testing has indicated that the distribution is limited to the range from 1.5 to 4.5. Write a KSL program to generate 10 observations from this distribution using stream 1.
Exercise 2.6 The interest rate for a capital project is unknown. An accountant has estimated that the minimum interest rate will between 2% and 5% within the next year. The accountant believes that any interest rate in this range is equally likely. You are tasked with generating interest rates for a cash flow analysis of the project. Write a KSL program to generate 10 observations from this distribution using stream 1.
Exercise 2.7 Consider the following probability density function:
\[f(x) = \begin{cases} \dfrac{3x^2}{2} & -1 \leq x \leq 1\\ 0 & \text{otherwise} \\ \end{cases}\]
Write a KSL program to generate 10 observations from this distribution using stream 1 using the inverse transform technique.
Exercise 2.8 Consider the following probability density function:
\[f(x) = \begin{cases} 0.5x - 1 & 2 \leq x \leq 4\\ 0 & \text{otherwise} \\ \end{cases}\]
Write a KSL program to generate 10 observations from this distribution using stream 1 using the inverse transform technique.
Exercise 2.9 Consider the following probability density function:
\[f(x) = \begin{cases} \dfrac{2x}{25} & 0 \leq x \leq 5\\ 0 & \text{otherwise} \\ \end{cases}\]
Write a KSL program to generate 10 observations from this distribution using stream 1 using the inverse transform technique.
Exercise 2.10 Consider the following probability density function:
\[f(x) = \begin{cases} \dfrac{2}{x^3} & x > 1\\ 0 & x \leq 1\\ \end{cases}\]
Write a KSL program to generate 10 observations from this distribution using stream 1 using the inverse transform technique.
Exercise 2.11 The times to failure for an automated production process have been found to be randomly distributed according to a Rayleigh distribution:
\[\ f(x) = \begin{cases} 2 \beta^{-2} x e^{(-(x/\beta)^2)} & x > 0\\ 0 & \text{otherwise} \end{cases}\]
Write a KSL program to generate 10 observations from this distribution using stream 1 using the inverse transform technique.
Exercise 2.12 Suppose that the processing time for a job consists of two distributions. There is a 30% chance that the processing time is lognormally distributed with a mean of 20 minutes and a standard deviation of 2 minutes, and a 70% chance that the time is uniformly distributed between 10 and 20 minutes.
Write a KSL program to generate 10 observations from this distribution using stream 1.
Exercise 2.13 Suppose that the service time for a patient consists of two distributions. There is a 25% chance that the service time is uniformly distributed with minimum of 20 minutes and a maximum of 25 minutes, and a 75% chance that the time is distributed according to a Weibull distribution with shape of 2 and a scale of 4.5.
Write a KSL program to generate 10 observations from this distribution using stream 1.
Exercise 2.14 Consider the following probability density function:
\[f(x) = \begin{cases} \dfrac{3x^2}{2} & -1 \leq x \leq 1\\ 0 & \text{otherwise} \\ \end{cases}\]
Write a KSL program to generate 10 observations from this distribution using the acceptance-rejection technique. Use stream 1 in your implementation.
Exercise 2.15 Consider the following function:
\[f(x) = \begin{cases} cx^{2} & a \leq x \leq b\\ 0 & \text{otherwise} \\ \end{cases} \]
- Determine the value of \(c\) that will turn \(g(x)\) into a probability density function. The resulting probability density function is called a parabolic distribution.
- Denote the probability density function found in part (a), \(f(x)\). Let \(X\) be a random variable from \(f(x)\). Derive the inverse cumulative distribution function for \(f(x)\).
- Write a KSL program to generate 10 observations from this distribution over the range of \(a=4\) and \(b=12\) using your work from part (a) and (b). Use stream 1 in your implementation.
Exercise 2.16 Consider the following probability density function:
\[f(x) = \begin{cases} \frac{3(c - x)^{2}}{c^{3}} & 0 \leq x \leq c\\ 0 & \text{otherwise} \\ \end{cases} \]
Derive an inverse cumulative distribution algorithm for generating from \(f(x)\). Write a KSL program to generate 10 observations from this distribution for \(c=5\). Use stream 1 in your implementation.