To make a truncated normal distribution in Python, you can use the `scipy.stats`

module. Here is the step-by-step process:

**Import the required libraries**: import numpy as np import scipy.stats as stats**Define the desired mean (mu) and standard deviation (sigma) for the normal distribution**: mu = 5 sigma = 2**Set the lower and upper limits for the truncation**: lower = 2 upper = 8**Calculate the z-scores corresponding to the lower and upper truncation limits**: lower_z = (lower - mu) / sigma upper_z = (upper - mu) / sigma**Create a truncated normal distribution using the stats.truncnorm function**: truncated_dist = stats.truncnorm((lower_z - mu) / sigma, (upper_z - mu) / sigma, loc=mu, scale=sigma) Note: The loc parameter is the mean and scale is the standard deviation.**Generate random numbers from the truncated normal distribution using the rvs method**: random_numbers = truncated_dist.rvs(size=1000) This will generate 1000 random numbers following the truncated normal distribution within the specified limits.

You can change the values of `mu`

, `sigma`

, `lower`

, `upper`

, and `size`

to customize the truncated normal distribution according to your needs.

## How to define the truncation limits for a truncated normal distribution?

The truncation limits for a truncated normal distribution can be defined based on the desired range of values for the distribution.

To define the truncation limits, you need to specify a lower and an upper bound for the distribution. These limits will restrict the range of values that the truncated normal distribution can take.

Here are the steps to define the truncation limits for a truncated normal distribution:

**Determine the lower truncation limit**: Decide on the minimum value that you want the distribution to be truncated at. This can be any real number or negative infinity (−∞).**Determine the upper truncation limit**: Decide on the maximum value that you want the distribution to be truncated at. This can be any real number or positive infinity (+∞).

It's important to note that the truncation limits should make sense in the context of the application or problem you are working on. For example, if you are interested in modeling heights of adults, you might set the lower truncation limit at 0 (since heights cannot be negative) and the upper truncation limit at a reasonable maximum height value.

Once you have determined the truncation limits, you can generate a truncated normal distribution with those limits using appropriate statistical software or programming languages.

## How to calculate the variance of a truncated normal distribution?

To calculate the variance of a truncated normal distribution, you can follow these steps:

- Determine the mean and standard deviation of the original normal distribution. Let's denote them as μ and σ, respectively.
- Identify the truncation points of the distribution, denoted as a (lower truncation point) and b (upper truncation point). These points define the interval within which the distribution is truncated.
- Calculate the cumulative distribution function (CDF) of the original normal distribution at points a and b. Let's denote these values as F(a) and F(b), respectively. This gives you the probability that the random variable falls below a or above b.
- Calculate the adjusted mean and standard deviation of the truncated normal distribution. The adjusted mean, denoted as μ', is given by:

μ' = μ + φ(a) - φ(b) / [F(b) - F(a)]

where φ(.) denotes the probability density function (PDF) of the standard normal distribution.

The adjusted standard deviation, denoted as σ', is given by:

σ' = σ * {1 - [φ(a) - φ(b)][F(b) - F(a)] / [f(a) - f(b)]}

where f(.) denotes the PDF of the original normal distribution.

- Finally, calculate the variance of the truncated normal distribution using the adjusted mean and standard deviation. The variance is given by:

Var = σ'^2

Note that the calculation of the variance for a truncated normal distribution involves adjusting the mean and standard deviation based on the truncation points and probabilities associated with them.

## What are some applications of truncated normal distribution?

There are several applications of truncated normal distribution, including:

**Financial modeling**: Truncated normal distribution is commonly used in finance for modeling asset returns, as it captures the restriction of prices or returns being bounded within certain limits.**Environmental modeling**: Truncated normal distribution is used in environmental science to model data that have underlying physical constraints, such as pollutant concentrations or species abundance, where negative values are not possible.**Risk analysis**: Truncated normal distribution is utilized in risk analysis to model uncertain variables, such as insurance claims or loss distributions, while considering lower and upper limits defined by the nature of the system.**Quality control**: Truncated normal distribution is applied in statistical process control to model manufacturing variables where measurement values fall within certain acceptable ranges and have known bounds.**Health research**: Truncated normal distribution is used in health research to model patient outcomes or measurements that have natural limits, such as blood pressure, body weight, or cholesterol levels.**Survival analysis**: Truncated normal distribution is employed in survival analysis, specifically when survival time is only known within a certain range, such as medical studies where follow-up periods are limited.**Educational testing**: Truncated normal distribution is used in educational testing to model test scores, as scores are often bounded within a specific range (e.g., 0 to 100) and cannot extend beyond the boundaries.

These are just a few examples, and there are many other fields where truncated normal distribution finds applications whenever data or variables are bound by specific limits.

## What is the mean of a truncated normal distribution?

The mean of a truncated normal distribution refers to the average value of the distribution. In a truncated normal distribution, some values that fall outside a specific range are excluded or "truncated," resulting in a narrower distribution. The mean of a truncated normal distribution will depend on the specific range or bounds used for truncation.

## How to calculate percentiles of a truncated normal distribution?

To calculate percentiles of a truncated normal distribution, you can follow these steps:

- Determine the parameters of the truncated normal distribution. This includes the mean (μ), standard deviation (σ), lower truncation point (a), and upper truncation point (b).
**Standardize the lower truncation point (a) and upper truncation point (b) using the standard normal distribution formula**: Z = (x - μ) / σ, where Z is the standardized value and x is the original value.- Calculate the cumulative distribution function (CDF) values for the standardized lower truncation point (Za) and upper truncation point (Zb) using a standard normal distribution table or a statistical software. The CDF represents the probability of getting a value less than or equal to a certain value.
**Calculate the percentile corresponding to a specific value (x) within the truncated normal distribution by using the formula**: percentile = (CDF(x) - CDF(a)) / (CDF(b) - CDF(a)) * 100.

For example, let's say we have a truncated normal distribution with a mean of 50, standard deviation of 10, lower truncation point of 40, and upper truncation point of 60. We want to calculate the 75th percentile.

**Parameters**: μ = 50, σ = 10, a = 40, b = 60.**Standardize**: Za = (40 - 50) / 10 = -1, Zb = (60 - 50) / 10 = 1.**Calculate CDF**: CDF(-1) = 0.1587, CDF(1) = 0.8413.**Calculate percentile**: percentile = (CDF(75) - CDF(-1)) / (CDF(1) - CDF(-1)) * 100.

This process allows you to calculate the desired percentile for a truncated normal distribution.

## How to sample from a truncated normal distribution in Python?

You can sample from a truncated normal distribution in Python using the `scipy.stats.truncnorm`

module. Here's an example code snippet:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 |
import numpy as np from scipy.stats import truncnorm # Define the truncation range a, b = -1, 1 # Define the mean and standard deviation mu, sigma = 0, 1 # Create a truncated normal distribution object dist = truncnorm((a - mu) / sigma, (b - mu) / sigma, loc=mu, scale=sigma) # Generate a random sample of size n n = 100 sample = dist.rvs(n) # Print the sample print(sample) |

In this example, the truncation range is set to [-1, 1], and the mean and standard deviation are set to 0 and 1 respectively. You can adjust these parameters to fit your specific requirements. The `dist.rvs(n)`

function generates `n`

random samples from the truncated normal distribution.