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F71SM STATISTICAL METHODS Tutorial on Section 3 RANDOM VARIABLES 1. A discrete random variable X has probability mass function x012 f (x) 0.25 0.5 0.25 Find the mean, variance, and probability generating function of X. [E[X] = 1, Var[X] = 0.5, Pgf: G(t) = (1 + t)2/4] 2. A discrete random variable X has probability […]

F71SM Statistical Methods Heriot-Watt University Computer Lab 1: Data summaries, plots and functions in R 1. The data in this task refer to the amount of claims (in £1000) arising in a general insurance portfolio. You can find the data in the file claim amounts data.txt on Vision under Learning Materials → Labs. Read data

Discrete distributions Uniform on {1, 2, . . . , k}: parameter k a positive integer; X is the outcome in the situation in which all outcomes 1, 2, . . . , k are equally likely, so with probability 1/k F71SM STATISTICAL METHODS 4 SPECIAL DISTRIBUTIONS We study a series of distributions that have

F71SM STATISTICAL METHODS Tutorial on Section 4 SPECIAL DISTRIBUTIONS (discrete distributions) 1. Let X ∼ b(n,p) and Y ∼ b(n,1−p). Note that P(X ≥ k) = P(Y ≤ n−k). Hence, using NCST, find the probabilities that in a series of 20 independent, identical Bernoulli trials with P (success) = 0.6, we obtain (i) 12 or

################################################### # Computer Lab 4 – F71SM ################################################### ## In the tasks below, fill in any missing code as ## required. ## Make sure you use the help menus in R (e.g.”?mean” ## will open up a help window for “mean()”). ## In some cases, the numerical answers you will find ## may be slightly

F71SM STATISTICAL METHODS Tutorial on Section 2 PROBABILITY 1. (a) BywritingA∪B∪CasA∪DwhereD=B∪C,verifythat P(A∪B∪C) = P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C) (b) BywritingA∩B∩C asD∩C ,whereD=A∩B,verifythat P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B) 2. Consider three events A, B, C which are such that A and B are mutually exclusive, A and C are independent, B and C are independent, P(A)

F71SM STATISTICAL METHODS 1 DATA SUMMARY 1.1 Introduction Data: our data consist of a sample x = (x1, x2, . . . , xn) from some population (real or concep- tual); in the case that every member of the population has the same chance of being included in the sample we have a (simple) random

################################################### # Computer Lab 4 – F71SM ################################################### ## In the tasks below, fill in any missing code as ## required. ## Make sure you use the help menus in R (e.g.”?mean” ## will open up a help window for “mean()”). ## In some cases, the numerical answers you will find ## may be slightly

1 Resources The home page for R is at http://www.r-project.org/. This contains links to obtain the software itself, gives access to the wealth of online documentation, and gives details of further resources, e.g. books. There are some recommended online tutorials at http://www.macs.hw.ac.uk/actuarial/R/ including how to download and install R on your computer. Starting and quitting

F71SM STATISTICAL METHODS 3 RANDOM VARIABLES 3.1 Introduction: random variables, distribution functions, proba- bility mass and density functions Given a probability space (S, P ), a random variable X is a real-valued function defined on S, i.e.X:S→RX ⊆R.Werequirethatforeveryx∈RX,thesubset{ω:ω∈SandX(ω)≤x} of S is an event. Consider A ⊆ RX . What is the meaning of the symbol