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Revision Notes
Practice Questions
Topical Past Paper Questions
Past Papers
Further Probability & Statistics
/
Probability generating functions
Contents
Constructing PGFs from a process
Model repeating trials and alternative branches before simplifying.
Practise Probability generating functions
Contents
All chapters
Further Mathematics · 9231
Further Probability & Statistics
Chapter 05 · Probability generating functions
Combining variables
Constructing PGFs from a process
01
Continuous random variables
Probability density
3
Understanding probability density
Checking and normalising a PDF
Piecewise PDFs and probabilities
Cumulative distributions
3
Building a cumulative distribution
Using and differentiating a CDF
Medians, quartiles and percentiles
Averages and spread
3
Finding the mode
Mean, variance and standard deviation
Expectations of functions of X
Transformed variables
2
Increasing transformations
Decreasing transformations and branches
Infinite ranges and review
2
Distributions on infinite intervals
Formula reference and mixed review
02
Inferential statistics
Sampling and t distributions
2
Sampling error and variance estimates
The t distribution and critical values
Hypothesis tests
4
Testing one population mean
Independent means and pooled variance
Paired sample t tests
Choosing the appropriate test
Confidence intervals
3
Confidence intervals for one mean
Confidence intervals for differences
Interval width, formula reference and review
03
Chi-squared tests
Goodness of fit
3
Observed counts, expected counts and discrepancy
Goodness-of-fit tests
Combining categories and degrees of freedom
Fitting distributions
2
Fitting binomial and Poisson distributions
Fitting continuous distributions
Independence
2
Independence in contingency tables
Formula reference and mixed review
04
Non-parametric tests
Sign and rank tests
3
Choosing a non-parametric test
Single-sample sign tests
Wilcoxon signed-rank tests
Comparing populations
2
Paired sign and signed-rank tests
Wilcoxon rank-sum tests
Normal approximations
3
Normal approximations for signs and signed ranks
Normal approximation for rank sums
Formula reference and mixed review
05
Probability generating functions
Constructing PGFs
3
What a probability generating function represents
Uniform and binomial PGFs
Geometric and Poisson PGFs
Using PGFs
2
Extracting probabilities from a PGF
Mean, variance and unknown probabilities
Combining variables
4
Sums of independent random variables
Transformations and repeated observations
Constructing PGFs from a process
Formula reference and mixed review
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