Explain what is meant by a discrete random variable, define the distribution function and the probability function of such a variable

Aim 

The aim of the Actuarial Statistics 1 subject is to provide a grounding in mathematical and statistical techniques that are of particular relevance to actuarial work.

Competences

On successful completion of this subject, a student will be able to: 

1   describe the essential features of statistical distributions. 

2   summarise data using appropriate statistical analysis, descriptive statistics and graphical presentation.

 3   describe and apply the principles of statistical inference. 

4   describe, apply and interpret the results of the linear regression model and generalised linear models.

5   explain the fundamental concepts of Bayesian statistics and use them to compute Bayesian estimators.

Links to other subjects 

CS2 – Risk Modelling and Survival Analysis builds directly on the material in this subject. 

CM1 – Actuarial Mathematics 1 and CM2 – Financial Engineering and Loss Reserving apply the material in this subject to actuarial and financial modeling. 

This subject assumes that a student will be competent in the following elements of Foundational Mathematics and basic statistics: 

1 Summarise the main features of a data set (exploratory data analysis) 

1.1  Summarise a set of data using a table or frequency distribution, and display it graphically using a line plot, a box plot, a bar chart, histogram, stem and leaf plot, or other appropriate elementary device.

1.3 Describe the spread/variability of a set of data using the standard deviation, range, interquartile range, as appropriate.

1.4 Explain what is meant by symmetry and skewness for the distribution of a set of data. 

2  Probability

2.1 Set functions and sample spaces for an experiment and an event.

2.2 Probability as a set function on a collection of events and its basic properties.

2.3  Calculate probabilities of events in simple situations.

2.4  Derive and use the addition rule for the probability of the union of two events.

2.5 Define and calculate the conditional probability of one event given the occurrence of another event.

2.6 Derive and use Bayes’ Theorem for events. 

2.7  Define independence for two events, and calculate probabilities in situations involving independence.

3 Random variables

3.1 Explain what is meant by a discrete random variable, define the distribution function and the probability function of such a variable, and use these functions to calculate probabilities. 

3.2  Explain what is meant by a continuous random variable, define the distribution function and the probability density function of such a variable, and use these functions to calculate probabilities. 

3.3 Define the expected value of a function of a random variable, the mean, the variance, the standard deviation, the coefficient of skewness and the moments of a random variable, and calculate such quantities.

 3.4  Evaluate probabilities associated with distributions (by calculation or by referring to tables as appropriate).

 3.5Derive the distribution of a function of a random variable from the distribution of the random variable.

Syllabus topics

1 Random variables and distributions (20%)

2 Data analysis (15%)

3 Statistical inference (20%)

4 Regression theory and applications (30%)

5 Bayesian statistics (15%)

 The weightings are indicative of the approximate balance of the assessment of this subject between the main syllabus topics, averaged over a number of examination sessions.

 The weightings also have a correspondence with the amount of learning material underlying each syllabus topic. However, this will also reflect aspects such as:

 ·       the relative complexity of each topic, and hence the amount of explanation and support required for it.

 ·       the need to provide thorough foundation understanding on which to build the other objectives.

 ·       the extent of prior knowledge which is expected 

·       the degree to which each topic area is more knowledge or application based.

 Skill levels 

The use of a specific command verb within a syllabus objective does not indicate that this is the only form of question which can be asked on the topic covered by that objective. The Examiners may ask a question on any syllabus topic using any of the agreed command verbs, as are defined in the document “Command verbs used in the Associate and Fellowship written examinations”. 

Questions may be set at any skill level: Knowledge (demonstration of a detailed knowledge and understanding of the topic), Application (demonstration of an ability to apply the principles underlying the topic within a given context) and Higher Order (demonstration of an ability to perform deeper analysis and assessment of situations, including forming judgements, taking into account different points of view, comparing and contrasting situations, suggesting possible solutions and actions, and making recommendations).