Question

Gaussian (Normal) Distribution. Function randomo returns a uniformly distributed random variable in the range \([0,1)\), which means that there is an equal probability of any given number in the range occurring on a given call to the function. Another type of random distribution is the Gaussian distribution, in which the random value takes on the classic bellshaped curve shown in Figure 5.9. A Gaussian distribution with an average of \(0.0\) and a standard deviation of \(1.0\) is called a standardized normal distribution, and the probability of any given value occurring in the standardized normal distribution is given by the equation $$ p(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} $$ It is possible to generate a random variable with a standardized normal distribution starting from a random variable with a uniform distribution in the range \([-1,1)\) as follows. 1\. Select two uniform random variables \(x_{1}\) and \(x_{2}\) from the range \([-1,1)\) such that \(x_{1}^{2}+x_{2}^{2}<1\). To do this, generate two uniform random variables in the range \([-1,1)\), and see if the sum of their squares happens to be less than \(1.0\). If so, use them. If not, try again. 2\. Then each of the values \(y_{1}\) and \(y_{2}\) in the equations that follow will be a normally distributed random variable. $$ \begin{aligned} &y_{1}=\sqrt{\frac{-2 \ln r}{r}} x_{1} \\ &y_{2}=\sqrt{\frac{-2 \ln r}{r} x_{2}} \end{aligned} $$ where $$ r=x_{1}^{2}+x_{2}^{2} $$ a In is the natural logarithm (log to the base e). Write a function that returns a normally distributed random value cach time that it is called. Test your function by getting 1000 random values, calculating the standard deviation, and plotting a histogram of the distribution. How close to \(1.0\) was the standard deviation?

Short answer

To solve this problem, we implement a function that returns normally distributed random values. We generate two uniform random variables, \(x_1\) and \(x_2\), in the range \([-1, 1)\) using the Python `random.uniform` function and ensure their sum of squares is less than 1.0. Then, we calculate the values of \(y_1\) and \(y_2\) using the derived equations, which are normally distributed random values. To test the function, we obtain 1000 random values, calculate their standard deviation using Python and matplotlib, and plot a histogram of the distribution. The calculated standard deviation should be close to 1.0, as the random values are expected to follow a standardized normal distribution.

Step-by-step solution

Generate uniform random variables in the range \([-1, 1)\)

To generate uniform random variables in the range \([-1, 1)\), we can use the random.uniform Python function. Keep generating \(x_1\) and \(x_2\) until their sum of squares is less than 1. ```python import random def generate_uniform_random_variable(): while True: x1 = random.uniform(-1, 1) x2 = random.uniform(-1, 1) if x1**2 + x2**2 < 1: return x1, x2 ```

Calculate \(y_1\) and \(y_2\)

Using the generated uniform random variables, we can now calculate \(y_1\) and \(y_2\). ```python import math def generate_normal(): x1, x2 = generate_uniform_random_variable() r = x1**2 + x2**2 y1 = math.sqrt((-2 * math.log(r)) / r) * x1 y2 = math.sqrt((-2 * math.log(r)) / r) * x2 return y1, y2 ```

Test the function

Generate 1000 random values from our function, calculate their standard deviation, and plot a histogram of the distribution. ```python import matplotlib.pyplot as plt # Generate 1000 random values random_values = [] for _ in range(500): y1, y2 = generate_normal() random_values.append(y1) random_values.append(y2) # Calculate standard deviation mean = sum(random_values) / len(random_values) squared_sum = sum((x - mean)**2 for x in random_values) standard_deviation = math.sqrt(squared_sum / len(random_values)) print("Standard Deviation:", standard_deviation) # Plot histogram plt.hist(random_values, bins=20, density=True) plt.xlabel('Value') plt.ylabel('Frequency') plt.title('Normal Distribution') plt.show() ``` This code will print the calculated standard deviation and show a histogram of the distribution of 1000 generated random values. The standard deviation should be close to 1.0, as the random values are expected to follow a standardized normal distribution.

Key concepts

MATLAB Random Variable Generation

Generating random variables in MATLAB is a fundamental aspect of simulations and stochastic analysis. The software provides several built-in functions to create random data according to different distributions, including the Gaussian (Normal) distribution. For example, to generate a random variable from a uniform distribution, MATLAB uses the rand function, which by default gives values between 0 and 1. To obtain values from a different range, one can simply scale and shift the output accordingly.

To adapt uniform random variables to follow a particular distribution, such as the Gaussian distribution, mathematical transformations are applied. In MATLAB, the Box-Muller transform is a common method used to convert uniformly distributed random variables into normally distributed ones. The process involves generating pairs of independent, standard normally distributed random variables using simple algebraic formulae and logarithms, leveraging the inherent features of the uniform distribution.

Steps to Generate Normal Random Variables in MATLAB

• Generate two uniform random variables.
• Check if they fit the criteria (their squared sum is less than 1).
• Apply the Box-Muller transformation to achieve normal distribution.
• Use MATLAB's vectorization features to generate large sets of random variables efficiently.

Standardized Normal Distribution

The standardized normal distribution, or the standard normal distribution, is a special case of the Gaussian distribution with a mean of 0 and a standard deviation of 1. This distribution is symmetrical and follows the bell curve shape that's characteristic of the Gaussian distribution. Its probability density function (PDF) is described by the equation \( p(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} \).

Understanding the properties of the standard normal distribution is crucial for statistical analysis since any normal distribution can be converted to a standard normal distribution using z-scores. This conversion can be done by subtracting the mean from the observed value and then dividing by the standard deviation. MATLAB's built-in functions normrnd or randn can directly generate random variables from a standardized normal distribution, thus simplifying simulation studies.

Key Characteristics of Standard Normal Distribution

• Has a mean of 0.
• Has a standard deviation of 1.
• The total area under the curve of the PDF is 1.
• Is fundamental in statistical z-tests and in generating random samples for various applications.

Histogram Plotting

Histograms are graphical representations of the distribution of numerical data, and they are particularly useful for visualizing the shape and spread of continuous sample data. In MATLAB, histograms can be plotted using the hist or histogram functions. When plotting the histogram of a normally distributed data set, the histogram will approximate the bell curve, with most data points concentrated around the mean and fewer as they move away towards the tails.

To display a histogram, MATLAB counts the number of data points that fall within a range of values, called bins. The height of each bar in the histogram corresponds to the count of data points within that bin's range. When plotted for a large sample size, the histogram provides a clear visualization of the theoretical probability density function on which the Gaussian distribution is based.

Practical Tips for Histogram Plotting in MATLAB

• Choose an appropriate number of bins for clarity.
• Ensure a large enough sample size for an accurate representation of the distribution.
• The histogram function has features for normalization to fit the probability density function.

Probability Density Function

The probability density function (PDF) is a statistical expression that describes the relative likelihood for a random variable to take on a given value. In the context of the Gaussian distribution, the PDF is bell-shaped and peaks at the mean, indicating the highest probability of occurrence. It mathematically assigns a probability to every possible value of the random variable within the distribution’s range.

For a standardized normal distribution, the PDF is defined by the equation \( p(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} \). This function is essential in various fields such as finance, engineering, and science, for it allows the calculation of the probability that a stochastic variable lies within a particular range of values. MATLAB can effortlessly work with PDFs of many distributions using the Statistics and Machine Learning Toolbox, which provides functions like normpdf for computing the PDF of a normal distribution.

Importance of PDF in Analysis

• Helps to determine probabilities and make decisions based on statistical models.
• Can be used to fit theoretical distributions to empirical data.
• Forms the foundation for hypothesis testing and confidence interval calculations.