A Level Mathematics S1 Revision Notes

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A Level Mathematics S1 Revision Notes

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > A Level Mathematics S1 Revision Notes

1 Representing Data
2 Probability
3 Discrete Random Variables
- 3.1 Discrete Uniform Distribution
4 Continuous Random Variables
5 The Normal Distribution
6 Correlation
7 Regression
8 Estimating and Samples

Representing Data

Probability

The probability, p, of something is the likelihood of that event occurring. $0 \leq p \leq 1$ where a probability of 0 means the event is impossible and a probability of 1 means the event is guaranteed to happen.

The sample space is every single possible outcome, while an event is a set of possible outcomes.

A Venn diagram shows the sample space, and a Tree diagram shows the events.

$\mathrm{P}(A \cap B)$ is the probability of A and B.

$\mathrm{P}(A \cup B)$ is the probability of A or B or both.

$\mathrm{P}(A|B)$ is the probability of A given that B has already happened.

$\mathrm{P}(A')$ is the probability of A not occurring.

$\mathrm{P}(A) + \mathrm{P}(A') = 1$

$\mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B) = \mathrm{P}(A \cup B)$ (Addition Rule)

$\mathrm{P}(A|B) \times \mathrm{P}(B) = \mathrm{P}(A \cap B)$ (Multiplication Rule)

If 2 events are independent, then $\mathrm{P}(A) \times \mathrm{P}(B) = \mathrm{P}(A \cap B)$ .

If 2 events are mutually exclusive, then $\mathrm{P}(A \cap B) = 0$

Discrete Random Variables

Discrete Random Variables are ones which can only take certain values, and not the values in between. A probability function describes the probability of each outcome, and a probability distribution is a table of all the outcomes along with their probabilities. A cumulative distribution function is in the form F(x), and is the probability of all the values of outcomes up to and including x.

The sum of all probabilities must equal 1.

$\mathrm{E}(X) = \displaystyle\sum_{\forall x} x\mathrm{P}(X = x)$

$\mathrm{E}(aX + b) = a\mathrm{E}(X) + b$

$\mathrm{Var}(X) = \mathrm{E}(X^2) - \mathrm{E}(X)^2$

$\mathrm{Var}(aX + b) = a^2\mathrm{Var}(X)$

Discrete Uniform Distribution

Not necessary, but saves time in an exam. A discrete uniform distribution is when all outcomes have an equal probability of occurring (e.g. a die roll). For a discrete uniform distribution:

$\mathrm{E}(X) = \dfrac{n + 1}{2}$

$\mathrm{Var}(X) = \dfrac{n^2 - 1}{12}$

Continuous Random Variables

The Normal Distribution

The probability distribution of a continuous random variable is represented by a curve; the area under the curve in a given interval gives the probability of a value lying in that interval.

If X is normally distributed with mean µ and standard deviation σ, then X~N(µ,σ^2)

If Z is a continuous random variable, where Z~N(0,1) then Φ(z) = P(Z<z)

The variable Z=(X-μ)/σ is the standard normal variable corresponding to X.

The percentage points table shows, for a probability p, the value of z such that P(Z<z) = p

Normal Distribution Graph

Correlation

2 variables are positively correlated if one increases with the other, and negatively correlated if one decreases as the other increases. The variables are usually plotted on a scatter diagram. Correlation is measured by the Product Moment Correlation Coefficient (PMCC), r, where:

$r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$

where:

$S_{xy} = \sum (x_i - \overline{x})(y_i - \overline{y}) = \sum x_iy_i - \dfrac{\sum x_i \sum y_i}{n}$

$S_{xx} = \sum (x_i - \overline{x})^2 = \sum x^2_i - \dfrac{(\sum x_i)^2}{n}$

$S_{yy} = \sum (y_i - \overline{y})^2 = \sum y^2_i - \dfrac{(\sum y_i)^2}{n}$

x_i/y_i = Each individual value of x/y,

$\overline{x}/\overline{y}$ = Mean of the x/y values

It is also possible to code the x and y values. If $u = \dfrac{x - a}{b}$ and $v = \dfrac{y - c}{d}$ , then $r = \dfrac{S_{uv}}{\sqrt{S_{uu}S_{vv}}}$

Regression

The independent/explanatory variable, usually x, is the one which can be set, while the dependent/response variable, y is the one which depends on the values of x.

Linear regression can be calculated by the least squares regression line y = a + bx , where:

$b = \dfrac{S_{xx}}{S_{yy}}$

$a = \overline{y} - b\overline{x}$

Interpolation is using the regression line to estimate y given x, while extrapolation is using the line to estimate y with a value of x outside the range of values used for the line. Extrapolation is not very useful as you do not know if the trend continues outside your range of values.