A mathematical model represents real-world systems using deterministic equations based on physical laws, while a statistical model uses probabilistic methods to describe patterns and relationships in data.

A mathematical model uses mathematical equations and expressions to describe a system, process, or phenomenon based on theoretical principles or assumptions.

A statistical model uses probabilistic methods to describe relationships between variables based on observed data.

Good, but not always the statistical model uses probabilities?

Quantum physics uses mathematics to demonstrate probabilistic outcomes. And are not probability assessments based upon mathematics. Is not radioactive decomposition based on mathematical analysis? At some point we enter a gray area. Perhaps the question is ill proposed. Is there such a thing as pure math or pure statistics? And what is the point of a yes or no answer? Are not probabilistic models mathematical?

Yes, probability models are mathematical, but they lack the concept of mathematics that builds itself by not having random error.@

Mathematical models and statistical models are both used to represent relationships and make predictions about real-world phenomena, but they differ in their foundations, purposes, and methodologies. Here's a breakdown of their key differences:

### Mathematical Model:

1. **Definition**: A mathematical model uses mathematical expressions and equations to represent a system or phenomenon. It seeks to describe relationships between variables using deterministic relationships.

2. **Nature**: Typically deterministic, meaning that given a set of inputs, the output will always be the same. There is often a clear, predefined relationship (e.g., physical laws).

3. **Examples**: Equations of motion in physics (e.g., Newton's laws), population growth models (e.g., logistic growth), and financial models (e.g., Black-Scholes model).

4. **Purpose**: To provide insights into the underlying mechanisms of a system, simulate scenarios, and predict outcomes based on known relationships. It often focuses on understanding and explaining phenomena.

5. **Input/Output**: Inputs are usually specific variables, and the output is a calculated result based on those inputs through mathematical operations.

### Statistical Model:

1. **Definition**: A statistical model uses statistical methods to analyze data and infer relationships between variables. It often incorporates randomness and uncertainty, acknowledging that real-world data can be noisy and incomplete.

2. **Nature**: Typically probabilistic, meaning it deals with the likelihood of various outcomes given certain inputs. It accounts for variability and uncertainty in the data.

3. **Examples**: Linear regression models, logistic regression models, time series analysis, and Bayesian models.

4. **Purpose**: To make inferences about populations based on sample data, estimate relationships between variables, and predict future observations. It often focuses on understanding data patterns and making decisions based on likelihoods.

5. **Input/Output**: Inputs are typically data points or observations, and the output is an estimate of relationships or probabilities, often with associated confidence intervals or error metrics.

### Summary:

In essence, mathematical models often seek to explain and predict based on established relationships and theoretical principles, while statistical models focus on analyzing data to uncover patterns and make probabilistic inferences. The choice between the two depends on the nature of the problem, the type of data available, and the goals of the analysis.

Good, but we cannot say with certainty that nature is inevitable.@

Follower@

النموذج الإحصائي هو تصنيف لنموذج رياضي، يجسّد مجموعة من الفرضيات الإحصائية فيما يتعلق بتكوين عينة

النماذج الإحصائية غير حتمية، أي أن المخرجات لا يتم تحديدها بالكامل بالمواصفات، وبالتالي فإن نفس المدخلات يمكن أن تنتج نتائج مختلفة لعمليات تشغيل مختلفة. النماذج الرياضية حتمية وستنتج دائمًا نفس المخرجات إذا كانت الظروف الأولية والحدودية متماثلة.

شكرا جزيلا دكتور

In a mathematical model, probabilities are known a priori, but not in a statistical model.

Following@

mathematics contains statistics as a chapter, so a model can be mathematical without being statistical.