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Chapter 5: Problem 7

Dice Simulation. It is often useful to be able to simulate the throw of a fair die. Write a MATLAB function dice that simulates the throw of a fair die by returning some random integer between 1 and 6 every time that it is called. (Hint: Call randomo to generate a randem number. Divide the possible values out of random0 into six equal intervals, and return the number of the interval that a given random value falls into.)

Short Answer

Expert verified
To create a MATLAB function called 'dice' that simulates the throw of a fair die, use the 'rand' function to generate a random number between 0 and 1 and divide the range into six equal intervals. Then, determine which interval the random number falls into and return the corresponding integer value (1 to 6). ```MATLAB function result = dice() random_number = rand; if random_number <= 1/6 result = 1; elseif random_number <= 2/6 result = 2; elseif random_number <= 3/6 result = 3; elseif random_number <= 4/6 result = 4; elseif random_number <= 5/6 result = 5; else result = 6; end end ```

Step by step solution

01

Create a MATLAB function

To create the 'dice' function, we will begin by adding the function signature and an end statement: ```MATLAB function result = dice() end ```
02

Generate a random number using 'rand' function

Now, we will generate a random number between 0 and 1 using the 'rand' function: ```MATLAB function result = dice() random_number = rand; end ```
03

Categorize the random number into one of the six intervals

Here, we will divide the range of random values generated by 'rand' into six equal intervals, and determine which interval the generated random number falls into: ```MATLAB function result = dice() random_number = rand; if random_number <= 1/6 result = 1; elseif random_number <= 2/6 result = 2; elseif random_number <= 3/6 result = 3; elseif random_number <= 4/6 result = 4; elseif random_number <= 5/6 result = 5; else result = 6; end end ``` Now, the 'dice' function is complete. Whenever the function is called, it will generate a random number between 0 and 1, classify it into one of the six equal intervals, and return the corresponding integer value (1 to 6), simulating the throw of a fair die.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

MATLAB programming
MATLAB stands out as a highly versatile tool used by engineers, scientists, and students to perform a vast array of computations, from simple arithmetic to complex numerical simulations. It鈥檚 designed to make matrix and vector operations straightforward and efficient, as well as providing high-level commands for data analysis, visualization, and developing algorithms.

In the context of our dice simulation exercise, MATLAB programming involves writing a function which is a self-contained block of code that accomplishes a specific task鈥攊n this case, simulating the roll of a die. The function is created with inputs and outputs defined, and can be called upon multiple times within a script or command window, which is a fundamental concept of programming to execute repeated tasks efficiently.
Random number generation
Within MATLAB, generating random numbers is a foundational element in simulations and stochastic processes. The 'rand' function is the most straightforward method and returns a pseudo-random number between 0 and 1. These values are uniformly distributed, meaning every number in this range is equally likely to occur.

For our dice simulation, we use this uniform distribution as our basis to simulate each possible outcome of a dice roll, which also has equal likelihood. This core concept is significant in many applications beyond our die-throw simulation, such as Monte Carlo methods, statistical sampling, and random process simulations.
Conditional statements
Conditional statements in MATLAB, using 'if', 'elseif', and 'else', allow for decision-making processes within a function or script. These statements execute code based on the evaluation of one or more conditions; if a condition is true, the corresponding block of code within the statement is executed.

In our die simulation, conditional statements are used to categorize the generated random number into one of six equally likely outcomes. By checking the random number against defined intervals, we simulate the sides (1 through 6) of a fair die. Mastery of conditional statements is crucial for creating interactive and responsive programs.
Function creation
Creating functions in MATLAB is an essential skill for organizing and managing your code. Functions are blocks of code that perform specific tasks and can be reused and called upon. When creating a function, it is given a name (in our case, 'dice'), and can have inputs and outputs. The construction of a function includes the function signature, the body which contains the executable code, and the end statement.

Through function creation in our exercise, we encapsulate the entire process of generating a random number and mapping it to a die's side into a neat, callable structure. This modularity is beneficial for simplifying complex tasks and improving code readability and maintainability.

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Most popular questions from this chapter

Read Traffic Density. Function random0 produces a number with a uniform probability distribution in the range \([0.0,1.0)\). This function is suitable for simulating random events if each outcome has an equal probability of occurring. However, in many events the probability of occurrence is not equal for every event, and a uniform probability distribution is not suitable for simulating such events. For example, when traffic engineers studied the number of cars passing a given location in a time interval of length \(t\), they discovered that the probability of \(k\) cars passing during the interval is given by the equation $$ P(k, t)=e^{-\lambda t} \frac{(\lambda t)^{4}}{k !} \text { for } t \geq 0, \lambda>0, \text { and } k=0,1,2, \ldots $$ This probability distribution is known as the Poisson distribution: it occurs in many applications in science and engineering. For example, the number of calls \(k\) to a telephone switchboard in time interval \(t\), the number of bacteria \(k\) in a specified volume \(t\) of liquid, and the number of failures \(k\) of a complicated system in time interval \(t\) all have Poisson distributions. Write a function to evaluate the Poisson distribution for any \(k, t\), and A. Test your function by calculating the probability of \(0,1,2, \ldots, 5\) cars passing a particular point on a highway in I minute, given that \(\lambda\) is \(1.6\) per minute for that highway. Plot the Poisson distribution for \(t=1\) and \(\lambda=1.6\).

Sort with Carry. It is often useful to sort an array are1 into ascending order, while simultancously carrying along a second array arr2. In such a sort, each time an element of array arrl is exchanged with another element of arr1, the corresponding elements of array arr2 are also swapped. When the sort is over, the elements of array arri are in ascending order, whereas the elements of array arr 2 that were associated with particular elements of array arrl are still associated with them. For example, suppose we have the following two arrays. \(\begin{array}{ccc}\text { Element } & \text { arr1 } & \frac{\operatorname{arc} 2}{1 .} \\ 1 . & 6 . & 0 . \\ 2 . & 1 . & 10 . \\\ 3 . & 2 . & 10 .\end{array}\) After sorting array arrl while carrying along array arr2, the contents of the two arrays will be \(\begin{array}{ccc}\text { Element } & \text { arr1 } & \text { are2 } \\ 1 . & 1 . & 0 . \\ 2 . & 2 . & 10 . \\ 3 . & 6 . & 1 .\end{array}\) Write a function to sort one real array into ascending order while carrying along a second one. Test the function with the following two 9 -element arrays. $$ \begin{aligned} &a=[-1, \quad 12, \quad-6, \quad 17, \quad-23,0,5, \quad 1,-1]: \\ &b=[-31,101,36,-17, \quad 0,10,-8,-1,-1]= \end{aligned} $$

Gravitational Force. The gravitational force \(F\) between two bodies of masses \(m_{1}\) and \(m_{2}\) is given by the equation $$ F=\frac{G m_{1} m_{2}}{r^{2}} $$ where \(G\) is the gravitational constant \(\left(6.672 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2} / \mathrm{kg}^{2}\right), m_{1}\) and \(m_{2}\) are the masses of the bodies in kilograms, and \(r\) is the distance between the two bodies. Write a function to calculate the gravitational force between two bodies given their masses and the distance between them. Test your function by determining the force on an \(800-\mathrm{kg}\) satellite in orbit \(38,000 \mathrm{~km}\) above the Earth. (The mass of the Earth is \(5.98 \times 10^{24} \mathrm{~kg}\).)

What is the difference between a script file and a function?

Rayleigh Distribution. The Rayleigh distribution is another random number distribution that appears in many practical problems. A Rayleighdistributed random value can be created by taking the square root of the sum of the squares of two normally distributed random values. In other words, to generate a Rayleigh-distributed random value \(r\) get two normally distributed random values \(\left(n_{1}\right.\) and \(\left.n_{2}\right)\), and perform the following calculation. $$ r=\sqrt{n_{1}^{2}+n_{2}^{2}} $$ a. Create a function rayleigh \((\mathrm{n}, \mathrm{m})\) that returns an \(\mathrm{n} \mathrm{x} \mathrm{m}\) array of Rayleigh-distributed random numbers. If only one argument is supplied [rayleigh \((\mathrm{n})\) ], the function should return an \(\mathrm{n} \times \mathrm{n}\) array of Rayleigh- distributed random numbers. Be sure to design your function with input argument checking and with proper documentation for the MATLAB help system. b. Test your function by creating an array of 20,000 Rayleigh-distributed random values and plotting a histogram of the distribution. What does the distribution look like? c. Determine the mean and standard deviation of the Rayleigh distribution.
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