• A Level Mathematics S1 Revision Notes

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    (a)	For the class  5 – 9:

Lower class boundary is 4.5 Upper class boundary is 9.5 Class width = 5 Class midpoint - "x" = 7

   (b)   Histogram:

Frequency density = Frequency / Class width



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


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}}}


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}}}

Estimating and Samples

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