Question

Independent vs. dependent random numbers. Generate a sequence of \(N\) independent random variables with values 0 or 1 and print out this sequence without space between the numbers (i.e., as 001011010110111010\() .\) The next task is to generate random zeros and ones that are dependent. If the last generated number was 0 , the probability of generating a new 0 is \(p\) and a new 1 is \(1-p\). Conversely, if the last generated was 1, the probability of generating a new 1 is \(p\) and a new 0 is \(1-p\). Since the new value depends on the last one, we say the variables are dependent. Implement this algorithm in a function returning an array of \(N\) zeros and ones. Print out this array in the condense format as described above. Choose \(N=80\) and try the probabilities \(p=0.5, p=0.8\) and \(p=0.9\). Can you by visual inspection of the output characterize the differences between sequences of independent and dependent random variables? Name of program file: dependent_random_variables.py.

Short answer

Independent sequences are completely random, while dependent sequences show patterns, especially with higher \(p\) values.

Step-by-step solution

Generate Independent Random Variables

Start by generating a sequence of \(N\) independent random variables. In Python, you can use the `random` module to accomplish this. You need to generate a random value of either 0 or 1 for each element in the sequence, which can be done using the `randint(0,1)` function within a for loop that iterates \(N=80\) times. As each value is generated, append it to a list and then convert the list to a string without spaces to resemble the format of 001011010110111010 as you print it. This sequence should represent independent random variables because each value is generated without consideration for the previous value.

Prepare Function for Dependent Random Variables

Define a function to generate dependent random variables. This function should take \(N\) and \(p\) as parameters. Initialize the first number randomly using `randint(0,1)`. This will be the starting point for your dependent sequence.

Generate Dependent Random Variables

Using the initialized starting number, iteratively generate each subsequent number based on the probability \(p\). If the last generated number was 0, use `random()` combined with \(p\) to decide if the next number is 0 (with probability \(p\)) or 1 (with probability \(1-p\)). Conversely, if the last number was 1, generate the next number based on \(p\) probability of being 1. Ensure this is repeated \(N=80\) times, storing each result in a list.

Convert Dependent Sequence to String Format

Once the sequence for dependent variables is generated, convert it into a single string without spaces. Print this string to visually compare it with the independent sequence.

Experiment with Different p Values

Repeat the dependent sequence generation with different values of \(p = 0.5, 0.8, 0.9\). In each case, ensure the same process is followed as described in Step 3 and print out the generated sequences. Visually inspect the output for each \(p\) value to observe how dependence affects the sequence's structure. Consider aspects like block sizes of consecutive 0s or 1s and frequency of switches between them.

Key concepts

Independent Random Numbers

Independent random numbers are generated in such a manner that each number is produced without any influence from the previous numbers generated. This means the outcome of one number doesn't alter the probability of the outcome of the next number. In the context of the provided exercise, each number is equally likely to be 0 or 1. You can think of this like flipping a fair coin multiple times; each flip is independent of the last.

• Each number's probability is fixed.
• No link between consecutive numbers.
• Easy to simulate using the `randint` function in Python.

To generate a sequence of independent random numbers in Python, you can utilize the `random` module and iterate through a loop with the `randint(0, 1)` function.

Dependent Random Numbers

Dependent random numbers differ from independent ones because each number generated in the sequence is influenced by the previous number. As outlined in the exercise, the probability for generating the next number changes depending on whether the previous number was a 0 or a 1. This behavior introduces a chain-like dependency, somewhat akin to being influenced by past events.

• The probability depends on the last generated number.
• Sequences can exhibit longer runs or blocks of similar numbers.
• Requires additional logic to implement in Python.

In the given task, you adjust the probability using a provided `p` value, which dictates whether to continue the sequence with a 0 or change to a 1, and vice versa. This method introduces an interesting variability depending on the chosen `p` values.

Probability

Probability is a fundamental concept in random variables, governing the likelihood of certain outcomes. It ranges from 0 (impossible) to 1 (certain). When dealing with independent numbers, each value has an equal chance, typically 0.5 for either a 0 or 1, similar to 50% heads or tails in coin tossing.

• Determines chance of each outcome in random sequences.
• Varies between independent and dependent scenarios.
• Can be adjusted to show different patterns in output.

For dependent sequences, the probability `p` plays a crucial role. If the last number is 0, `p` indicates the chance of producing another 0 and vice versa for 1. Fine-tuning `p` can make one type of number more frequent, altering the structure of randomness seen in the sequence.

Python Programming

Python is an excellent language for working with random variables because of its simplicity and powerful libraries. In this exercise, the `random` module is key. It lets you produce random numbers effortlessly, which is essential for both independent and dependent sequences.

• Use `random.randint(0, 1)` for independent sequences.
• Adapt `random.random()` with logic for dependent sequences.
• Python loops and conditionals are vital for implementing the logic.

Starting with the basic framework, you can define functions that encapsulate random number logic, allowing for customization like different `p` values to observe changes. This reflects Python's flexibility and efficiency in handling complex probability-driven tasks in programming.