A discrete random variable takes countable, separate values, each with a probability. This chapter defines the probability distribution function for a discrete variable, shows how to find its expected value and standard deviation, and introduces the main discrete models: the binomial, geometric, hypergeometric, and Poisson distributions.
The Probability Distribution Function
A discrete random variable takes values that can be counted, such as the number of heads in ten coin tosses. Its probability distribution function lists each possible value together with the probability of that value occurring. Two conditions must hold: every probability lies between zero and one, and the probabilities of all outcomes add up to one. Presented as a table, the distribution gives a complete description of how likely each outcome is.
Expected Value and Standard Deviation
The expected value of a discrete random variable is its long-run average, found by multiplying each value by its probability and adding the results. It is the mean of the distribution and need not equal any single possible outcome. The standard deviation measures how much the values typically differ from the expected value, calculated from the squared distances of each value from the mean weighted by their probabilities. Together they summarise the centre and spread of the distribution.
Common Discrete Distributions
Several named distributions describe common situations. The binomial distribution counts successes in a fixed number of independent trials, each with the same probability of success. The geometric distribution counts the trials needed to get the first success. The hypergeometric distribution applies when sampling without replacement from two groups, and the Poisson distribution models the number of events in a fixed interval when they occur independently at a constant average rate.
Statistics begins with data and how it is gathered. This chapter defines the key terms of population, sample, parameter, and statistic, explains common sampling methods and why sampling well matters, and introduces frequency tables and levels of measurement used to organise data before any analysis begins.
Key Terms: Population and Sample
Statistics is the study of how to collect, organise, analyse, and interpret data. A population is the entire group being studied, while a sample is the smaller part of it actually observed, since measuring a whole population is rarely practical. A number that describes a population is called a parameter, and a number that describes a sample is called a statistic. Because samples stand in for populations, a statistic is used to estimate the parameter it corresponds to.
Sampling Methods
How a sample is chosen decides how well it represents the population. In simple random sampling every member has an equal chance of selection, while stratified sampling divides the population into groups and draws from each. Cluster sampling selects whole groups at random, and systematic sampling takes every nth member from a list. Convenience sampling uses whoever is easiest to reach, but it risks bias because the sample may not reflect the wider population.
Frequency Tables and Levels of Measurement
A frequency table records how often each value or class appears, and relative frequency expresses that count as a proportion of the total. Data is also classified by its level of measurement: nominal data names categories, ordinal data can be ordered but its gaps are not meaningful, interval data has meaningful gaps but no true zero, and ratio data has a true zero so ratios make sense. The level of measurement guides which calculations are appropriate.
Descriptive statistics organise and summarise data so patterns become clear. This chapter shows how to display data with stem-and-leaf plots, histograms, and box plots, then how to measure the centre of a data set using the mean, median, and mode, and its spread using the range, variance, and standard deviation.
Displaying Data Graphically
Once data is collected it must be organised before it can be understood. Graphs turn a long list of numbers into a shape the eye can read. Stem-and-leaf plots and line graphs keep the original values visible, histograms and frequency polygons show how often values occur across intervals, and box plots summarise the spread and highlight the median and quartiles. Choosing the right display depends on the data and the question being asked.
Measures of the Centre
A single value can represent where a data set is centred. The mean is the arithmetic average, found by adding the values and dividing by how many there are. The median is the middle value once the data is ordered, and it is less affected by extreme values. The mode is the value that occurs most often. When a distribution is skewed, the mean is pulled toward the longer tail, so the median often describes typical values better.
Measures of Spread and Location
Spread describes how far the data is scattered around its centre. The range is the difference between the largest and smallest values, while the variance and its square root, the standard deviation, measure the average distance of values from the mean. Location measures such as quartiles and percentiles mark the points below which a given fraction of the data falls, and together these summarise both the width and the shape of a data set.
A continuous random variable is measured, not counted, so probability is found as area under a probability density function, not at single points. This chapter defines continuous probability functions, then develops two models: the uniform distribution, where outcomes are equally likely, and the exponential distribution, used for waiting times and decay.
Probability as Area Under a Curve
A continuous random variable is measured rather than counted, so its values fall across an interval instead of at separate points. Its behaviour is described by a probability density function, written f(x), whose graph is a curve. Probability is represented by the area between that curve and the x-axis, and the entire area under the curve equals one. Because probability is area, it is defined for intervals of x rather than for any single value, so the probability that x equals one exact point is zero.
The Uniform Distribution
In a uniform distribution every outcome in the range is equally likely, and the density function is a horizontal line of constant height. For a variable spread evenly between a and b, f(x) equals 1/(b − a) across that interval, and probabilities are found as the area of a rectangle: base times height. For example, with f(x) = 1/20 on the interval 0 to 20, the probability that x lies between 0 and 2 is (2 − 0)(1/20) = 0.1. The distribution is written X ~ U(a, b).
The Exponential Distribution
The exponential distribution models the time between events, such as how long a phone call lasts or a car battery survives, and is written X ~ Exp(m), where m is the decay parameter. Its density function is f(x) = m·e^(−m·x), and the cumulative distribution function P(X < x) = 1 − e^(−m·x) gives probability as area up to a point. Unlike the uniform distribution, smaller values are more likely than larger ones, so the curve falls away from left to right. Worked examples in the chapter use it to find warranty periods and expected lifetimes.
Time series: observations of well-defined data at regular intervals (weekly/monthly/yearly). Components: trend (long-term), cyclical (1-7 years), seasonal (yearly), irregular (random). Models: multiplicative (Y=T×C×S×I), additive (Y=T+C+S+I). Moving average smoothing for forecasting.
