Mathematical Models Flashcards
What is the role of statistical tests in mathematical modelling?
To assess how well a mathematical model matches a real world situation
What are the 7 stages of mathematical modelling?
- Real world problem observed
- Mathematical model devised
- Predictions made
- Experimental data collected
- Predictions compared to outcomes
- Statistical test assesses success of model
- Model is refined
Advantages of mathematical models?
- Quick and easy to produce
- They can simplify a more complex situation
- They enable predictions to be made about the future
Disadvantages of mathematic models?
- They only give a partial description of the real situation
* They only work for a restricted range of values
Continuous variable
A variable which can take any value in a given range
Discrete variable
A variable that can only take values within a given range
How to find median in discrete data?
Divide n by 2
How to find the median in continuous data?
Divide n by 2, and then use interpolation to find the exact term
What is the height of a histogram bar?
Frequency density
What is frequency density?
Frequency ÷ class width
Which data are histograms used to display?
Continuous
Ways to describe skew of data?
•Shape from box plots
• 3(mean-median)
————————
Standard deviation
P(A’)=…
= 1- P(A)
P (A U B) =…
= P(A) + P(B) - P(A ∩ B)
and vice versa
P(A|B) =…
= P(A∩B)
————
P(B)
P(A∩B)=…
P(A|B) x P(B) or P(B|A) x P(A)
When are A and B independent?
P(A|B) = P(A)
or
P(A∩B) = P(A) x P(B)
When are A and B mutually exclusive?
P(A∩B) = 0
What does a represent in a regression line?
The point at which y touches the x axis
What does b represent on in a regression line?
The amount by which y increase for an increase of 1 in x (gradient)
What is an independent/explanatory variable?
One which is set independently of the other variable
What is a dependent/response variable?
One whose value are determined by the values of the independent variable
How to calculate b in the regression line?
Sxy
———
Sxx
P(A’)=…
= 1- P(A)
P (A U B) =…
= P(A) + P(B) - P(A ∩ B)
and vice versa
P(A|B) =…
= P(A∩B)
————
P(B)
P(A∩B)=…
P(A|B) x P(B) or P(B|A) x P(A)
When are A and B independent?
P(A|B) = P(A)
or
P(A∩B) = P(A) x P(B)
When are A and B mutually exclusive?
P(A∩B) = 0
What does a represent in a regression line?
The point at which y touches the x axis
What does b represent on in a regression line?
The amount by which y increase for an increase of 1 in x (gradient)
What is an independent/explanatory variable?
One which is set independently of the other variable
What is a dependent/response variable?
One whose value are determined by the values of the independent variable
How to calculate b in the regression line?
Sxy
———
Sxx
How to calculate a in the regression line?
a= (mean of y) - b( mean of x)
Interpolation
Estimate of the value of the dependent variable within the range of the data
Extrapolation
When you estimate a value outside the range of data
What is an important property of discrete random variables?
Sum of them =1
How to find F(x)?
Add together all the for all those outcomes which are less than or equal to x
What is E(X)?
ΣxP(X=x) or mean
What is Var(X)?
Var(X) = E(X^2) - E(X)^2
What are the conditions for a discrete uniform distribution?
- Each value is equally likely
* A dislike random variable X is defined over a set of n values
What is the E(X) for a discrete uniform distribution?
E(X) = n+1
——
2
where n is the final value of x
How to calculate Var(X) for a discrete uniform distribution?
(n+1)(n-1)
Var(X) = —————
12
where n is the final value of x
