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Further Mathematics · 9231

Further Probability & Statistics

01Continuous random variables

Probability density3

Understanding probability density Checking and normalising a PDF Piecewise PDFs and probabilities

Cumulative distributions3

Building a cumulative distribution Using and differentiating a CDF Medians, quartiles and percentiles

Averages and spread3

Finding the mode Mean, variance and standard deviation Expectations of functions of X

Transformed variables2

Increasing transformations Decreasing transformations and branches

Infinite ranges and review2

Distributions on infinite intervals Formula reference and mixed review

02Inferential statistics

Sampling and t distributions2

Sampling error and variance estimates The t distribution and critical values

Hypothesis tests4

Testing one population mean Independent means and pooled variance Paired sample t tests Choosing the appropriate test

Confidence intervals3

Confidence intervals for one mean Confidence intervals for differences Interval width, formula reference and review

03Chi-squared tests

Goodness of fit3

Observed counts, expected counts and discrepancy Goodness-of-fit tests Combining categories and degrees of freedom

Fitting distributions2

Fitting binomial and Poisson distributions Fitting continuous distributions

Independence2

Independence in contingency tables Formula reference and mixed review

04Non-parametric tests

Sign and rank tests3

Choosing a non-parametric test Single-sample sign tests Wilcoxon signed-rank tests

Comparing populations2

Paired sign and signed-rank tests Wilcoxon rank-sum tests

Normal approximations3

Normal approximations for signs and signed ranks Normal approximation for rank sums Formula reference and mixed review

05Probability generating functions

Constructing PGFs3

What a probability generating function represents Uniform and binomial PGFs Geometric and Poisson PGFs

Using PGFs2

Extracting probabilities from a PGF Mean, variance and unknown probabilities

Combining variables4

Sums of independent random variables Transformations and repeated observations Constructing PGFs from a process Formula reference and mixed review

Revision Notes Practice Questions Topical Past Paper Questions Past Papers

Resources

Revision Notes Practice Questions Topical Past Paper Questions Past Papers

Further Probability & Statistics/Probability generating functions

Sums of independent random variables

Connect joint outcomes with multiplication of PGFs.

Practise Probability generating functions

Contents

Further Mathematics · 9231

Further Probability & Statistics

01Continuous random variables

Probability density3

Understanding probability density Checking and normalising a PDF Piecewise PDFs and probabilities

Cumulative distributions3

Building a cumulative distribution Using and differentiating a CDF Medians, quartiles and percentiles

Averages and spread3

Finding the mode Mean, variance and standard deviation Expectations of functions of X

Transformed variables2

Increasing transformations Decreasing transformations and branches

Infinite ranges and review2

Distributions on infinite intervals Formula reference and mixed review

02Inferential statistics

Sampling and t distributions2

Sampling error and variance estimates The t distribution and critical values

Hypothesis tests4

Testing one population mean Independent means and pooled variance Paired sample t tests Choosing the appropriate test

Confidence intervals3

Confidence intervals for one mean Confidence intervals for differences Interval width, formula reference and review

03Chi-squared tests

Goodness of fit3

Observed counts, expected counts and discrepancy Goodness-of-fit tests Combining categories and degrees of freedom

Fitting distributions2

Fitting binomial and Poisson distributions Fitting continuous distributions

Independence2

Independence in contingency tables Formula reference and mixed review

04Non-parametric tests

Sign and rank tests3

Choosing a non-parametric test Single-sample sign tests Wilcoxon signed-rank tests

Comparing populations2

Paired sign and signed-rank tests Wilcoxon rank-sum tests

Normal approximations3

Normal approximations for signs and signed ranks Normal approximation for rank sums Formula reference and mixed review

05Probability generating functions

Constructing PGFs3

What a probability generating function represents Uniform and binomial PGFs Geometric and Poisson PGFs

Using PGFs2

Extracting probabilities from a PGF Mean, variance and unknown probabilities

Combining variables4

Sums of independent random variables Transformations and repeated observations Constructing PGFs from a process Formula reference and mixed review

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