S1 summary

Chapter 1 summary - Mathematical modelling in probability and statistics

Chapter 2 summary - Representation of sample data

1. For a stem and leaf diagram each row represents a stem and is indicated by teh number to the left of the vertical line. The digits to the right of the vertical line are the leaves associated with the stem.
   
   
2. A grouped frequency distribution consists of several classes and their associated class frequencies.
   
   For the class 5-9 for example the

   lower class boundary	is	4.5
   lower class limit	is	5
   upper class limit	is	9
   upper class boundary	is	9.5
   class width	is	9.5 - 4.5 = 5
   class mid-point	is	{1\over 2}(4.5+9.5) = 7

   
   
3. When drawing a histogram, for each histogram bar the area is directly proportional to the frequency that it is representing:
   
   Area \propto Frequency
   
   and since the histogram consists of a series of bars, then for a histogram:
   
   Total Area \propto Total Frequency
   
   
4. The height of a histogram bar is found by dividing the class frequency by the class width.
   
   
5. Histograms are plotted using class boundaries.

Chapter 3 summary - Methods for summarising sample data (location)

1. The mode is that value of a variate which occurs most frequently.
   
   
2. The median is the middle value of an ordered set of data.
   
   
3. The quartiles of an ordered set of data are such that 25% of the observations are less than or equal to the first quartile (Q_1), 50% are less than or equal to the second quartile (Q_2) and 75% are less than or equal to the third quartile (Q_3).
   
   
4. The mean of a set of observations is the sum of all the observations divided by the total number of observations, i.e.
   
   \displaystyle \mu = \bar x = {\sum x\over n} \qquad or \displaystyle \qquad {\sum fx\over \sum f}

Chapter 4 summary - Methods for summarising data (dispersion)

1. The range of a data set is given by:
   
   Range = largest value - smallest value
   
   
2. The interquartile range is given by
   
   {\rm IQR} = Q_3 - Q_1
   
   
3. The semi-interquartile range is defined as:
   
   {\rm SIQR} = {1 \over 2}(Q_3 - Q_1)
   
   
4. Variance of a population is defined as:
   
   \displaystyle \sigma^2 = {\sum (x-\mu)^2\over n} \qquad or \displaystyle \qquad \sigma^2 = {\sum f(x-\mu)^2\over \sum f}
   
   
5. Unbiased estimator of the population variance is defined as:
   
   \displaystyle s^2 = {\sum(x-\bar x)^2\over n-1} \qquad or \displaystyle s^2 = {\sum f(x-\bar x)^2\over \sum f - 1}
   
   
6. The standard deviation is the positive square root of the variance.
   
   
7. For

   positive skew:	Q_2 - Q_1 \quad < \quad Q_3 - Q_2
   negative skew:	Q_2 - Q_1 \quad > \quad Q_3 - Q_2
   symmetry:	Q_2 - Q_1 \quad = \quad Q_3 - Q_2

Chapter 5 summary - Probability

1. {\rm P}(\text {event } A \text { or event } B) = {\rm P}(A \cup B)
   {\rm P}(\text {both events } A \text { and } B) = {\rm P}(A \cap B)
   {\rm P}(\text {not event } A) = {\rm P}(A')
   
   
2. Complementary probability
   
   {\rm P}(A') = 1 - {\rm P}(A)
   
   
3. Addition rule
   
   {\rm P}(A \cup B) = {\rm P}(A) + {\rm P}(B) - {\rm P}(A \cap B)
   
   
4. Conditional probability
   
   \displaystyle {\rm P}(A \text { given } B) = {\rm P}(A|B) = {{\rm P}(A \cap B)\over {\rm P}(B)}
   
   
5. Multiplication rule
   
   \displaystyle {\rm P}(A \cap B) = {\rm P}(A|B) \times {\rm P}(B)
   
   
6. A and B are independent events if
   
   {\rm P}(A \cap B) = {\rm P}(A) \times {\rm P}(B)
   
   
7. A and B are mutually exclusive events if
   
   {\rm P}(A \cap B) = 0

Chapter 6 summary - Correlation

1. Product-moment correlation coefficient:
   
   \displaystyle r = {S_{xy}\over \sqrt {S_{xx}S_yy}}
   
   where
   
   \displaystyle S_{xy} = \sum (x_i - \bar x)(y_i - \bar y) = \sum x_iy_i - {\sum x_i \sum y\over n}
   
   \displaystyle S_{xx} = \sum (x_i - \bar x)^2 = \sum x_i^2 - {(\sum x_i)^2\over n}
   
   \displaystyle S_{yy} = \sum (y_i - \bar y)^2 = \sum y_i^2 - {(\sum y_i)^2\over n}
   
   
2. r is a measure of linear association
   
   r = \quad 1 \Rightarrow perfect positive linear correlation
   r = -1 \Rightarrow perfect negative linear correlation
   r = \quad 0 \Rightarrow no linear correlation

Chapter 7 summary - Regression

1. Explanatory or independent variable:
   a variable that is set independently of the other variable
   
   
2. Response or dependent variable:
   the variable whose values are decided by the values of the explanatory or independent variable.
   
   
3. Linear regression model:
   
   y_1 = \alpha + \beta x_i + \varepsilon_i
   
   
4. The regression line of y on x is:
   
   y = a + bx,
   
   where \displaystyle \qquad b = {S_{xy}\over S_{xx}} \qquad and \qquad a = \bar y - b\bar x

Chapter 8 summary - Discrete random variables

1. For a discrete random variable X
   
   \displaystyle \sum_{\forall x}^{\quad} {\rm P}(X=x) = 1
   
   \displaystyle \mu = {\rm E}(X) = \sum_{\forall x}^{\text { }} x{\rm P}(X=x)
   
   \displaystyle \sigma^2 = {\rm E}(X^2) - \mu^2 = \sum_{\forall x}^{\text { }} x^2{\rm P}(X=x) - \mu^2
   
   
2. Properties of expected values and variance
   
   {\rm E}(aX + b) = a{\rm E}(X) + b
   
   {\rm Var}(aX + b) = a^2{\rm Var}(X)
   
   
3. Cumulative distribution function {\rm F}(x)
   
   0 \leq {\rm F}(x) \leq 1
   
   
4. For the discrete random variable X:
   
   \displaystyle {\rm F}(x_0) = {\rm P}(X \leq x_0) = \sum_{x \leq x_0} {\rm P}(X=x)

Chapter 9 summary - The normal distribution

1. For a continuous random variable X, having a normal distribution,
   
   Mean = \mu
   Variance = \sigma^2
   
   
2. Given that X \sim {\rm N}(\mu, \sigma^2),
   
   then \displaystyle \quad Z = {X - \mu\over \sigma} \sim {\rm N}(0,1^2)