Statistical Distributions: Evaluate and analyze standard discrete and continuous distributions, calculate probabilities, and fit distributions to observation

Did you know that statistical distributions play a crucial role in data analysis and decision making? Understanding the properties of different distributions can help you in making accurate predictions and drawing meaningful insights from your data.

📊 The single task "Statistical Distributions: Evaluate and analyze standard discrete and continuous distributions, calculate probabilities, and fit distributions to observed data" involves working with different types of distributions, including Binomial, Poisson, Normal, Log Normal, and Exponential, and performing various statistical analyses on them.

✨ Let's dive deeper into the different aspects of this task:

📈 Analyze the Statistical Distribution of a Discrete Random Variable

A discrete random variable is a variable that can only take on a finite or countably infinite number of values. Examples of discrete random variables include the number of defective products in a batch or the number of customers who visit a store on a given day.

To analyze the statistical distribution of a discrete random variable, you need to first determine its probability mass function (PMF), which gives the probability of each possible outcome. You can then use this PMF to calculate various statistics, such as the mean, variance, and standard deviation.

Here's an example of how to calculate the PMF and mean of a Binomial distribution in R:

# Calculate the PMF of a Binomial distribution

n <- 10

p <- 0.5

x <- 0:n

pmf <- dbinom(x, n, p)

# Calculate the mean of the Binomial distribution

mean <- sum(x * pmf)

📊 Calculate Probabilities using R for Binomial and Poisson Distributions

The Binomial and Poisson distributions are commonly used to model count data. The Binomial distribution is used when the number of trials is fixed and the outcomes are independent, while the Poisson distribution is used when the number of events in a fixed interval of time or space follows a Poisson process.

To calculate probabilities using R for these distributions, you can use the dbinom and dpois functions, respectively. These functions take in the values of the random variable, the number of trials/events, and the probability/rate parameter, and return the probability of getting that value.

Here's an example of how to calculate the probability of getting exactly 3 heads in 5 coin tosses using a Binomial distribution in R:

# Calculate the probability of getting exactly 3 heads in 5 coin tosses

n <- 5

p <- 0.5

x <- 3

prob <- dbinom(x, n, p)

📈 Fit Binomial and Poisson Distributions to Observed Data

In real-world scenarios, you may have observed data that you want to fit to a Binomial or Poisson distribution to analyze its properties. To do this, you can use the fitdistr function in R, which fits a distribution to the observed data and returns the estimated parameters.

Here's an example of how to fit a Poisson distribution to observed data in R:

# Fit a Poisson distribution to observed data

observed_data <- c(2, 3, 1, 4, 2, 0, 5, 3, 2)

fit <- fitdistr(observed_data, "poisson")

📊 Evaluate the Properties of Normal and Log Normal Distributions

The Normal and Log Normal distributions are continuous distributions that are commonly used to model continuous data. The Normal distribution is symmetric and has a bell-shaped curve, while the Log Normal distribution is skewed and has a long tail towards higher values.

To evaluate the properties of these distributions, you need to determine their probability density function (PDF), which gives the probability of observing a value within a certain range. You can then use this PDF to calculate various statistics, such as the mean, variance, and standard deviation.

Here's an example of how to calculate the PDF and mean of a Normal distribution in R:

# Calculate the PDF of a Normal distribution

mu <- 0

sigma <- 1

x <- seq(-3, 3, by = 0.1)

pdf <- dnorm(x, mean = mu, sd = sigma)

# Calculate the mean of the Normal distribution

mean <- mean(x)

📊 Calculate Probabilities using R for Normal and Log Normal Distributions

To calculate probabilities using R for Normal and Log Normal distributions, you can use the dnorm and dlnorm functions, respectively. These functions take in the values of the random variable, the mean and standard deviation/parameters, and return the probability of getting that value.

Here's an example of how to calculate the probability of observing a value between 0 and 1 in a Normal distribution with mean 0 and standard deviation 1 in R:

# Calculate the probability of observing a value between 0 and 1 in a Normal distribution

mu <- 0

sigma <- 1

prob <- pnorm(1, mean = mu, sd = sigma) - pnorm(0, mean = mu, sd = sigma)

📈 Fit Normal, Log Normal, and Exponential Distributions to Observed Data

Similar to fitting Binomial and Poisson distributions, you can also fit Normal, Log Normal, and Exponential distributions to observed data using the fitdistr function in R.

Here's an example of how to fit a Normal distribution to observed data in R:

# Fit a Normal distribution to observed data

observed_data <- rnorm(100, mean = 0, sd = 1)

fit <- fitdistr(observed_data, "normal")

📊 Evaluate the Concept of Sampling Distribution (t, F, and Chi Square)

The sampling distribution is the probability distribution of a statistic based on a random sample from a population. The t, F, and Chi Square distributions are commonly used in hypothesis testing and confidence interval estimation.

To evaluate the concept of sampling distribution, you need to understand how these distributions are related to the population parameters and how they are used in statistical inference.

Here's an example of how to calculate a t statistic in R:

# Calculate a t statistic for a two-sample t-test

x <- rnorm(50, mean = 10, sd = 2)

y <- rnorm(50, mean = 12, sd = 2)

t_stat <- t.test(x, y)$statistic

📊 Formulate Research Hypotheses and Perform Hypothesis Testing

Formulating research hypotheses and performing hypothesis testing is a crucial part of statistical inference. Hypothesis testing involves making a claim about a population parameter and testing whether the observed data supports or contradicts that claim.

To perform hypothesis testing, you need to first formulate the null and alternative hypotheses and then choose an appropriate statistical test based on the type of data and research question. You can then perform the test using R or Python programs and interpret the results.

Here's an example of how to perform a one-sample t-test in R:

# Perform a one-sample t-test

x <- rnorm(50, mean = 10, sd = 2)

t_test <- t.test(x, mu = 9)

p_value <- t_test$p.value

📊 Analyze the Concept of Variance (ANOVA) and Select an Appropriate ANOVA or ANCOVA Model

ANOVA (Analysis of Variance) is a statistical technique used to investigate differences between two or more groups. ANCOVA (Analysis of Covariance) is a variation of ANOVA that takes into account a covariate that may be affecting the outcome variable.

To analyze the concept of variance and select an appropriate ANOVA or ANCOVA model, you need to first define the variables, factors, and levels for your research problem. You can then evaluate the sources of variation and define a linear model for ANOVA/ANCOVA. Finally, you need to confirm the validity of assumptions and perform the analysis using R or Python programs.

Here's an example of how to perform a one-way ANOVA in R:

# Perform a one-way ANOVA

x <- rnorm(50, mean = 10, sd = 2)

y <- rnorm(50, mean = 12, sd = 2)

z <- rnorm(50, mean = 15, sd = 2)

anova <- aov(c(x, y, z) ~ c(rep("Group 1", 50), rep("Group 2", 50), rep("Group 3", 50)))

summary(anova)

By mastering these skills, you'll be able to analyze and interpret various types of data and draw meaningful insights that can inform your decision making.

Evaluate the statistical distribution of a discrete random variable using its probability mass function and cumulative distribution function.

Probability Mass Function and Cumulative Distribution Function in Discrete Random Variables

In the world of statistics, understanding the behavior of discrete random variables is essential for various applications. For instance, when dealing with the number of defective items in a batch, or the number of goals scored in a soccer match, we essentially work with discrete random variables. In this context, probability mass function (PMF) and cumulative distribution function (CDF) are two vital tools to evaluate the statistical distribution of such variables.

The Role of Probability Mass Function

A probability mass function is a function that assigns probabilities to discrete outcomes of a random variable. For a discrete random variable X, the probability mass function can be denoted as P(X = x), where x represents the possible values that X can take.

Let's explore this concept further with an example:

Imagine you're running an ice cream shop, and you want to analyze the sales data. You have the daily records of how many scoops of ice cream you sold in a week. The data is as follows:

Monday: 5 scoops

Tuesday: 3 scoops

Wednesday: 7 scoops

Thursday: 2 scoops

Friday: 5 scoops

Saturday: 8 scoops

Sunday: 4 scoops

To evaluate the statistical distribution, let's first create a probability mass function using the frequency of each possible value of scoops sold:

Scoops (X):     2   3   4   5   7   8

Frequency (f):  1   1   1   2   1   1

Probability (P): 1/7 1/7 1/7 2/7 1/7 1/7

The PMF table above shows the probability of selling a specific number of scoops each day. For example, the probability of selling 5 scoops in a day is 2/7.

Understanding Cumulative Distribution Function

A cumulative distribution function is a function that calculates the probability of a random variable being less than or equal to a given value. For a discrete random variable X, the CDF can be denoted as F(x) = P(X ≤ x).

Continuing with the ice cream shop example, let's calculate the cumulative distribution function from the probability mass function:

Scoops (X):     2   3   4   5   7   8

Probability (P): 1/7 1/7 1/7 2/7 1/7 1/7

CDF (F(x)):    1/7 2/7 3/7 5/7 6/7 7/7

The CDF table above shows the probability of selling less than or equal to a certain number of scoops in a day. For instance, the probability of selling less than or equal to 5 scoops is 5/7.

Practical Applications

Evaluating the statistical distribution of discrete random variables using PMF and CDF can help in various practical applications, such as:

• Inventory Management: By analyzing the demand for a specific product, companies can optimize their inventory and avoid stockouts or overstocking.

• Quality Control: By evaluating the distribution of defective items, manufacturers can identify potential issues in the production process and implement corrective measures.

• Risk Assessment: Insurance companies can use the distribution of claims to price their policies and manage their risk exposure.

In summary, understanding the statistical distribution of discrete random variables through probability mass function and cumulative distribution function is essential for making data-driven decisions in various fields. By analyzing the frequency of specific outcomes and their cumulative probabilities, businesses and researchers can gain valuable insights and improve their decision-making processes.

Calculate probabilities for Binomial and Poisson Distributions using R by specifying the number of trials, probability of success, and the number of events.

Calculating Probabilities for Binomial and Poisson Distributions using R

To calculate probabilities for Binomial and Poisson distributions using R, you should understand the basic concepts of these distributions and know how to use R functions for probability calculations. Let's dive into the details of each distribution and learn how to calculate probabilities using R.

Binomial Distribution: An Overview 📊

The Binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success. A Bernoulli trial is a random experiment with only two possible outcomes: success (1) or failure (0). Some examples of Bernoulli trials are coin tosses and product defect tests.

To calculate probabilities using the Binomial distribution, you need to know:

• Number of trials (n)

• Probability of success in each trial (p)

• Number of successful events (k)

The probability mass function (PMF) for the Binomial distribution is given by:

P(X=k) = C(n,k) * p^k * (1-p)^(n-k)

Poisson Distribution: An Overview 📈

The Poisson distribution is another discrete probability distribution that describes the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence (λ). Examples of events following the Poisson distribution include the number of phone calls received at a call center and the number of customers arriving at a store.

To calculate probabilities using the Poisson distribution, you need to know:

• Average rate of occurrence (λ)

• Number of events (k)

The probability mass function (PMF) for the Poisson distribution is given by:

P(X=k) = (λ^k * e^(-λ)) / k!

Calculating Probabilities for Binomial Distribution using R 📝

To calculate probabilities for the Binomial distribution in R, you can use the dbinom() function. The syntax for the function is:

dbinom(x, size, prob)

Where:

• x: Number of successful events (k)

• size: Number of trials (n)

• prob: Probability of success in each trial (p)

Example:

Suppose you want to calculate the probability of getting 3 heads in 5 coin tosses, where the probability of getting a head in each toss is 0.5. You can use the dbinom() function like this:

# Binomial probability calculation

probability <- dbinom(3, size = 5, prob = 0.5)

probability

Calculating Probabilities for Poisson Distribution using R 📝

To calculate probabilities for the Poisson distribution in R, you can use the dpois() function. The syntax for the function is:

dpois(x, lambda)

Where:

• x: Number of events (k)

• lambda: Average rate of occurrence (λ)

Example:

Suppose you want to calculate the probability of receiving 7 phone calls at a call center in an hour, given that the average rate of calls is 5 per hour. You can use the dpois() function like this:

# Poisson probability calculation

probability <- dpois(7, lambda = 5)

probability

Now that you have learned how to calculate probabilities for Binomial and Poisson distributions using R, you can apply these techniques to various real-world scenarios involving discrete probability distributions!

Fit Binomial and Poisson distributions to observed data using maximum likelihood estimation and evaluate goodness of fit using statistical tests.

Maximum Likelihood Estimation (MLE) for Binomial and Poisson Distributions

Maximum Likelihood Estimation (MLE) is a popular method to fit statistical distributions to observed data. The fundamental idea is to identify the parameters of the distribution that make the observed data most likely to occur. In this tutorial, we will discuss MLE for two popular discrete distributions: Binomial and Poisson. We will also evaluate the goodness of fit using statistical tests.

Fitting a Binomial Distribution Using MLE

The Binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of Bernoulli trials, each with the same probability of success. The distribution is characterized by two parameters: the number of trials (𝑛) and the probability of success (𝑝).

1. Collect the data: Gather a sufficient number of observations (successes and failures in each trial) from the process you want to model.

2. Calculate the sample mean and variance: Compute the sample mean (𝑥̅) and the sample variance (𝑠²) from the data. These will be used as the basis for estimating the parameters of the Binomial distribution.

import numpy as np

data = np.array([5, 7, 4, 8, 6, 5, 2, 9, 4, 7]) # sample data

sample_mean = np.mean(data)

sample_variance = np.var(data)

3. Estimate the parameters 𝑛 and 𝑝: Use the sample mean and variance to estimate the parameters 𝑛 and 𝑝.

𝑝 = 1 - (s² / x̅)

𝑛 = x̅ / p

estimated_p = 1 - (sample_variance / sample_mean)

estimated_n = sample_mean / estimated_p

4. Evaluate the goodness of fit: Perform a goodness of fit test, such as the Chi-squared test, to determine if the fitted Binomial distribution is a good model for the data.

Fitting a Poisson Distribution Using MLE

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given an average rate of occurrence (𝜆). The single parameter that characterizes the distribution is 𝜆.

1. Collect the data: Gather a sufficient number of observations (number of events) from the process you want to model.

2. Calculate the sample mean: Compute the sample mean (𝑥̅) from the data. This will be the basis for estimating the parameter of the Poisson distribution.

3. 
   

data = np.array([2, 3, 1, 4, 6, 2, 3, 5, 2, 7]) # sample data

sample_mean = np.mean(data)

3. Estimate the parameter 𝜆: Use the sample mean to estimate the parameter 𝜆.

𝜆 = x̅

estimated_lambda = sample_mean

4. Evaluate the goodness of fit: Perform a goodness of fit test, such as the Chi-squared test, to determine if the fitted Poisson distribution is a good model for the data.

Goodness of Fit Tests: The Chi-Squared Test

The Chi-squared test is a statistical test that helps us determine if the fitted distribution is a good model for the observed data. It compares the observed frequencies with the expected frequencies that would be obtained if the data followed the fitted distribution.

1. Calculate the expected frequencies: Compute the expected frequencies for each bin (range of values) using the fitted Binomial or Poisson distribution.

2. Compute the Chi-squared statistic: Calculate the Chi-squared statistic using the observed and expected frequencies.

𝜒² = Σ[(Oᵢ - Eᵢ)² / Eᵢ]

where Oᵢ are the observed frequencies and Eᵢ are the expected frequencies.

observed_frequencies = np.array([10, 15, 20, 25, 30]) # sample data

expected_frequencies = np.array([12, 14, 18, 24, 32]) # from fitted distribution

chi_squared = np.sum((observed_frequencies - expected_frequencies)**2 / expected_frequencies)

3. Determine the degrees of freedom: Calculate the degrees of freedom (df) for the test. For a Binomial or Poisson distribution, this is the number of bins minus one.

df = len(observed_frequencies) - 1

4. Compare the statistic with the critical value: Compare the computed Chi-squared statistic with the critical value from the Chi-squared distribution table for the given level of significance (e.g., 0.05) and degrees of freedom.

If the Chi-squared statistic is greater than the critical value, we reject the null hypothesis that the data follows the fitted distribution. If it is less than the critical value, we cannot reject the null hypothesis, and the fitted distribution is considered a good model for the data.

By following these steps, you can effectively fit Binomial and Poisson distributions to observed data using maximum likelihood estimation and evaluate the goodness of fit using statistical tests.

Analyze the properties of Normal and Log Normal distributions, including mean, variance, skewness, and kurtosis.

Analyzing Properties of Normal and Log Normal Distributions 📈

When it comes to statistical distributions, two of the most commonly encountered are the Normal and Log Normal distributions. These distributions have some fascinating properties that make them very useful in various applications in statistics, finance, and engineering. In this explanation, we'll dive deep into the properties of these distributions, namely their mean, variance, skewness, and kurtosis, and explore some real-world examples.

Properties of Normal Distribution 📊

The Normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve. It is defined by two parameters: the mean (μ) and the standard deviation (σ). The probability density function (PDF) of a Normal distribution is given by:

f(x) = (1 / (σ * √(2π))) * e^(-0.5 * ((x - μ) / σ)^2)

Mean (μ): The mean of a Normal distribution is the central point around which the data is symmetrically distributed. It represents the average value of the distribution.

Variance (σ^2): The variance of a Normal distribution measures the dispersion or spread of the data around the mean. The square root of the variance gives the standard deviation (σ), which is also a measure of spread.

Skewness (γ): Skewness measures the asymmetry of a distribution. For a Normal distribution, skewness is always 0, indicating perfect symmetry.

Kurtosis (κ): Kurtosis measures the "tailedness" or the concentration of values near the mean as compared to the tails. For a Normal distribution, kurtosis is 3, which indicates a mesokurtic distribution (neither too peaked nor too flat).

Properties of Log Normal Distribution 📉

The Log Normal distribution is a continuous probability distribution of a random variable whose logarithm follows a Normal distribution. If Y = ln(X) is normally distributed, then X will have a log-normal distribution. It is also defined by two parameters: the mean (μ) and the standard deviation (σ). The probability density function (PDF) of a Log Normal distribution is given by:

f(x) = (1 / (x * σ * √(2π))) * e^(-0.5 * ((ln(x) - μ) / σ)^2)

Mean (𝜇'): The mean of a Log Normal distribution is not the same as the mean of the underlying Normal distribution (μ). It can be calculated as 𝜇' = e^(μ + (σ^2 / 2)). This value represents the central tendency of the distribution.

Variance (σ'^2): The variance of a Log Normal distribution is different from the variance of the underlying Normal distribution (σ^2). It can be calculated as σ'^2 = (e^(σ^2) - 1) * e^(2μ + σ^2). This value measures the dispersion of the data.

Skewness (γ'): Skewness for a Log Normal distribution is always positive, which indicates that the distribution is positively skewed or right-skewed. It can be calculated as γ' = (e^(σ^2) + 2) * √(e^(σ^2) - 1).

Kurtosis (κ'): Kurtosis for a Log Normal distribution is always greater than 3, which indicates a leptokurtic distribution (more peaked than a Normal distribution). It can be calculated as κ' = e^(4σ^2) + 2 * e^(3σ^2) + 3 * e^(2σ^2) - 3.

Real-World Examples 🌎

Normal Distribution Example: Heights of People 👨‍👩‍👧‍👦

One of the most common real-world examples of a Normal distribution is the distribution of heights of people. For instance, if we measure the heights of a large number of adult men, we would find that the data is symmetrically distributed around the mean height, with most men having heights close to the mean and few men having extremely tall or short heights. The distribution would have a skewness of 0 and a kurtosis of 3.

Log Normal Distribution Example: Stock Prices 📈

Log Normal distributions are often used to model stock prices or other financial data. The reason for this is that stock prices cannot be negative, and the Log Normal distribution is always positive. Furthermore, stock prices tend to have a positive skew, with a greater likelihood of large increases than large decreases. This characteristic is captured well by the Log Normal distribution, which has a positive skewness and a kurtosis greater than 3.

In summary, understanding the properties of Normal and Log Normal distributions, such as their mean, variance, skewness, and kurtosis, is essential when working with statistical data. These distributions are widely used in various fields, making them valuable tools for any data analyst or statistician.

Calculate probabilities for Normal and Log Normal distributions using R by specifying the mean and standard deviation or the location and scale parameters.

Calculating Probabilities for Normal and Log-Normal Distributions in R 📊

Calculating probabilities for Normal and Log-Normal distributions is a common task in statistics, especially when analyzing real-world data. R is a powerful programming language that provides various built-in functions to work with these distributions. Let's dive into the details with some practical examples!

Normal Distribution: A Brief Overview 📚

A Normal Distribution, also known as a Gaussian distribution, is a continuous probability distribution, characterized by its bell-shaped curve. It is defined by two parameters: the mean (µ) and standard deviation (σ). The mean represents the central location of the distribution, while the standard deviation measures the spread of the distribution.

In R, the primary functions used for the Normal distribution are pnorm(), dnorm(), and qnorm():

• pnorm(): Calculates the cumulative distribution function (CDF), giving the probability that a value falls below a given point.

• dnorm(): Computes the probability density function (PDF), giving the probability density at a given point.

• qnorm(): Finds the quantile function, giving the value below which a specified proportion of the distribution lies.

• 
  

Log-Normal Distribution: A Brief Overview 📚

The Log-Normal Distribution is a continuous probability distribution of a random variable whose logarithm follows a Normal distribution. In other words, if Y is a Log-Normal distributed variable, then log(Y) has a Normal distribution. It is defined by two parameters: location (µ) and scale (σ).

In R, the primary functions for the Log-Normal distribution are plnorm(), dlnorm(), and qlnorm():

• plnorm(): Calculates the cumulative distribution function (CDF) for the Log-Normal distribution.

• dlnorm(): Computes the probability density function (PDF) for the Log-Normal distribution.

• qlnorm(): Finds the quantile function for the Log-Normal distribution.

• 
  

Example 1: Calculating Probabilities for the Normal Distribution in R 📚

Let's consider a scenario where the time taken by students to complete a test follows a Normal distribution with a mean (µ) of 60 minutes and a standard deviation (σ) of 10 minutes. We want to find the probability that a randomly selected student will finish the test in less than 45 minutes.

mu <- 60

sd <- 10

threshold <- 45

# Calculate the probability using pnorm()

prob <- pnorm(threshold, mean = mu, sd = sd)

prob

This will give us the probability that a student will finish the test in less than 45 minutes, which in this case is around 0.1587, or 15.87%.

Example 2: Calculating Probabilities for the Log-Normal Distribution in R 📚

Imagine we have a dataset of house prices, and we find that the distribution of these prices follows a Log-Normal distribution with a location (µ) of 2 and a scale (σ) of 0.5. We want to know the probability that a randomly selected house will have a price less than $10,000.

location <- 2

scale <- 0.5

threshold <- log(10000)

# Calculate the probability using plnorm()

prob <- plnorm(threshold, meanlog = location, sdlog = scale)

prob

This will give us the probability that a house will have a price less than $10,000, which in this case is around 0.0062, or 0.62%.

In conclusion, R offers various built-in functions to calculate probabilities for Normal and Log-Normal distributions, allowing you to analyze real-world data with ease. Familiarizing yourself with these functions and their applications will improve your statistical analysis skills and help you make informed decisions based on data. Happy analyzing! 🎉

Fit Normal, Log Normal, and Exponential distributions to observed data using maximum likelihood estimation and evaluate goodness of fit using statistical tests.Fitting Distributions to Observed Data

In the field of statistics, it's common to encounter data that doesn't initially follow any recognizable pattern or distribution. To make meaningful inferences from this data, it's important to fit it to a known distribution, such as Normal, Log Normal, or Exponential. Maximum Likelihood Estimation (MLE) is a powerful method for fitting these distributions to observed data.

📌 Maximum Likelihood Estimation (MLE): A statistical method used to estimate the parameters of a distribution by maximizing the likelihood function.

Fitting Normal, Log Normal, and Exponential Distributions

Before we dive into the details, let's first understand the three distributions we'll be working with:

• Normal Distribution: A continuous probability distribution characterized by its bell-shaped curve, also known as the Gaussian distribution.

• Log Normal Distribution: A continuous probability distribution of a random variable whose logarithm follows a normal distribution.

• Exponential Distribution: A continuous probability distribution that represents the time between events in a Poisson process.

Now let's explore how to fit these distributions to observed data using MLE.

Step 1: Calculate the MLEs of Parameters

To fit any of the aforementioned distributions to observed data, we must first calculate the MLEs of their parameters. For the Normal distribution, the parameters are μ (mean) and σ² (variance), while for the Log Normal distribution, the parameters are μ (mean) and σ² (variance) of the logarithm of the variable. The Exponential distribution has a single parameter λ (rate).

👩‍💻 Here's an example using Python to calculate MLEs for these parameters:

import numpy as np

import scipy.stats as stats

data = np.array([2, 4, 6, 8, 10])

# Normal Distribution

normal_mu = np.mean(data)

normal_sigma = np.std(data)

# Log Normal Distribution

log_data = np.log(data)

log_normal_mu = np.mean(log_data)

log_normal_sigma = np.std(log_data)

# Exponential Distribution

exp_lambda = 1/np.mean(data)

Step 2: Fit the Distributions to the Data

Now that we have the MLEs of the parameters, we can fit the distributions to our observed data.

👩‍💻 Continuing the Python example:

# Normal Distribution

normal_fit = stats.norm(loc=normal_mu, scale=normal_sigma)

# Log Normal Distribution

log_normal_fit = stats.lognorm(s=log_normal_sigma, scale=np.exp(log_normal_mu))

# Exponential Distribution

exp_fit = stats.expon(scale=1/exp_lambda)

Step 3: Evaluate Goodness of Fit Using Statistical Tests

After fitting the distributions, we need to evaluate their goodness of fit using statistical tests. The Kolmogorov-Smirnov (KS) test and the Anderson-Darling (AD) test are two common methods for comparing the goodness of fit of different distributions.

📌 Kolmogorov-Smirnov (KS) Test: A non-parametric test that compares the cumulative distribution function (CDF) of a sample with a reference probability distribution.

📌 Anderson-Darling (AD) Test: A statistical test used to compare the goodness of fit of a sample to a given distribution, placing more weight on the tails.

👩‍💻 Let's run these tests in Python:

# Kolmogorov-Smirnov Test

ks_normal = stats.kstest(data, normal_fit.cdf)

ks_log_normal = stats.kstest(data, log_normal_fit.cdf)

ks_exp = stats.kstest(data, exp_fit.cdf)

# Anderson-Darling Test

ad_normal = stats.anderson(data, dist='norm')

ad_log_normal = stats.anderson(np.log(data), dist='norm')

ad_exp = stats.anderson(data, dist='expon')

💡 The smaller the test statistic (D for KS test, A² for AD test) and the larger the p-value, the better the fit. Generally, if the p-value is greater than a significance level (e.g., 0.05), we can't reject the null hypothesis that the data follows the given distribution.

In conclusion, fitting Normal, Log Normal, and Exponential distributions to observed data can be achieved through the MLE method and evaluating their goodness of fit using statistical tests like the KS test and the AD test. This process can provide valuable insights into the underlying patterns and behaviors of the data, which is crucial for making informed decisions in various fields, such as finance, healthcare, and engineering.