1.1 Mathematical Models In Porbability And Statist
What is a mathematical model? A simplification of a real world situation. These are useful because:
- They are quick and easy to produce.
- They can simplify a more complex situation.
- Improve our understanding of the real world as certain variable can be changed.
- Allows predictions to be made.
- They can help provide control.
However, they should be treated with caution because:
- It is a simplification of the real problem so does not include all aspects of the problem.
- The model may only work in certain situations.
- Only work for a restricted range of values.
1.2 Designing a mathematical model
You can design a mathematical model by:
- 1. A real world problem is observed.
- 2. A mathematical model is devised.
- 3. The mathematical model is used to make predictions about the expected behaviour of the real world problem.
- 4. Experimental data is collected from the real world.
- 5. Compare predicted and observed outcomes.
- 6. Statistical tests are used to assess how well the model describes the real world.
- 7. The mathematical model is refined, if necessary, to improve the match of predicted outcomes with observed experimental data.
2.1 Representation and summary of data
Quantitative data: Numerical observations.
Qualitative data: Non-numerical observations.
Continuous Variable: Can take any value in a given range.
Discrete Data: Can only take specific values in a given range.
Oulier: An extreme value that lies outside the overall pattern of data. This is any value that is:
- Greater than upper quartile + (1.5 X IQR)
- Less than lower quartile - (1.5 X IQR)
- Q2-Q1 = Q3 -Q2 then the distribution is symmetrical.
- Q2-Q1 < Q3-Q2 then the distribution is positively skewed.
- Q2-Q1 > Q3-Q2 then the distribution is negatively skewed.
Can also use measures of location for skewness:
- mode=median=mean describes a distribution which is symmetrical.
- mode<median<mean describes a distribution with positive skewness.
- mode>median>mean describes a distribution with negative skewness.
You can calculate 3(mean - median) divided by standard deviation to find skewness.
- The larger the number the greater the amount of skewness.
- The closer to zero the more symmetrical the skewness is.
- Positive number means positive skewness, and negative number means negative skewness.
- Rule of addition: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- Rule of multiplication: P(A ∩ B) = P(A) P(B|A)
- Rule of subtraction: P(A') = 1 - P(A)
- P(A|B)=P(A n B)/P(B)
- Mutually Exclusive Events: P(A U B) = P(A)+P(B), where there are no outcomes in common or not overlap. P(A n B)=0
- When one event has no effect over another they are independant. To prove an independant event then P(A n B)=P(A) X P(B)