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Question 1Actuarial Statistics (CS1)

In a Markov chain with states {1,2,3}, the transition matrix has rows summing to 1. Which statement best describes a stationary distribution π?

  1. π satisfies Pπ = 0 with entries summing to 1
  2. π equals any row of the transition matrix P
  3. π satisfies πP = π with non-negative entries summing to 1 Correct answer.
  4. π must equal the uniform distribution over states

Explanation

A stationary distribution is a probability vector left-invariant under the transition matrix, so it satisfies πP = π and sums to one. It is not generally a row of P nor the uniform distribution.

Question 2Actuarial Statistics (CS1)

A random variable X has probability generating function G_X(t) = E(t^X). Which expression gives E[X]?

  1. G_X''(1)
  2. G_X'(1) Correct answer.
  3. G_X(0)
  4. G_X(1)

Explanation

Differentiating the pgf at t = 1 yields the first factorial moment, which equals E[X]. The second derivative gives the second factorial moment, and G_X(0) equals P(X = 0).

Question 3Actuarial Statistics (CS1)

Let N(t) be a Poisson process with rate λ. Given N(3) = 2, what is the probability that exactly one event occurred in the interval (0,1]?

  1. 1/2
  2. 2/9
  3. 1/4
  4. 4/9 Correct answer.

Explanation

Conditional on N(3)=2, the two event times are independently uniform on (0,3]. Each falls in (0,1] with probability 1/3, so the count is Binomial(2, 1/3). Thus P = 2·(1/3)·(2/3) = 4/9.

Question 4Actuarial Statistics (CS1)

For a compound Poisson model S = Σ_{i=1}^{N} X_i where N is Poisson(λ) independent of claim sizes X_i, which formula gives Var(S)?

  1. λ·E[X^2] Correct answer.
  2. λ·E[X]
  3. λ^2·E[X^2]
  4. λ·E[X]^2

Explanation

By the law of total variance, Var(S) = E[N]Var(X) + Var(N)(E[X])^2. With Var(N) = E[N] = λ, this simplifies to λ·E[X^2] by the second moment identity.

Question 5Actuarial Statistics (CS1)

A life table gives l_40 = 90,000 and l_41 = 89,100. What is the value of q_40, the probability a life aged 40 dies before age 41?

  1. 0.0100 Correct answer.
  2. 0.0111
  3. 0.9000
  4. 0.0090

Explanation

The one-year death probability is q_40 = (l_40 − l_41)/l_40. Substituting gives (90,000 − 89,100)/90,000 = 900/90,000 = 0.01.

Question 6Actuarial Statistics (CS1)

Under a constant force of mortality μ, which expression gives the survival probability ₜp_x for a life aged x?

  1. (1 − μ)^t
  2. exp(−μt) Correct answer.
  3. exp(−μ) · t
  4. 1 − μt

Explanation

A constant force of mortality implies the future lifetime is exponential with parameter μ, so ₜp_x = exp(−μt). This is the standard survival function of the exponential distribution.

Question 7Actuarial Statistics (CS1)

In a proportional hazards model h(t|x) = h_0(t)·exp(β^T x), what does a coefficient β_j = 0.7 imply for a one-unit increase in covariate x_j?

  1. The baseline hazard increases by 0.7
  2. The survival time increases by 0.7 units
  3. The hazard is multiplied by exp(0.7) Correct answer.
  4. The hazard is multiplied by 0.7

Explanation

The model is multiplicative in the linear predictor, so a one-unit change in x_j multiplies the hazard by exp(β_j). With β_j = 0.7, the hazard ratio is exp(0.7) ≈ 2.01.

Question 8Actuarial Statistics (CS1)

In a binomial model with n trials and unknown proportion p, which is the maximum likelihood estimator of p?

  1. X/(n + 1)
  2. (X + 1)/(n + 2)
  3. X/n Correct answer.
  4. 1/2

Explanation

The binomial log-likelihood is maximised where the derivative vanishes, giving the sample proportion p̂ = X/n. The other forms are shrinkage or ad hoc estimators, not the MLE.

Question 9Actuarial Statistics (CS1)

A statistic T is sufficient for a parameter θ if which condition holds?

  1. The distribution of the sample given T does not depend on θ Correct answer.
  2. T is unbiased for θ
  3. T has minimum variance among all estimators
  4. T is a linear function of the sample mean

Explanation

Sufficiency means the conditional distribution of the data given T is free of θ, so T captures all information about θ. Unbiasedness and minimum variance are separate properties.

Question 10Actuarial Statistics (CS1)

A sample of size n = 100 gives a sample mean of 50 and a known population variance of 400. What is the standard error of the sample mean?

  1. 2 Correct answer.
  2. 0.4
  3. 4
  4. 20

Explanation

The standard error is σ/√n = 20/10 = 2. The population standard deviation is √400 = 20, and dividing by √100 = 10 gives 2.

Question 11Actuarial Statistics (CS1)

A 95% confidence interval for a mean is constructed as x̄ ± 1.96·σ/√n with known σ. Which interpretation is correct?

  1. In repeated sampling, 95% of such intervals would contain μ Correct answer.
  2. 95% of the population values lie in the interval
  3. There is a 95% probability that μ lies in this specific interval
  4. The sample mean lies in the interval with probability 0.95

Explanation

Frequentist coverage refers to the long-run procedure: across repeated samples, 95% of constructed intervals capture the fixed parameter μ. A single realised interval either contains μ or not.

Question 12Actuarial Statistics (CS1)

In a hypothesis test, the p-value is defined as which quantity?

  1. The probability of a Type II error
  2. The probability the null hypothesis is true
  3. The probability of observing a test statistic at least as extreme as the observed, assuming the null is true Correct answer.
  4. The significance level chosen before the test

Explanation

The p-value is the tail probability of the test statistic under the null hypothesis, measuring evidence against it. It is not the probability that the null is true, nor the pre-set significance level.

Question 13Actuarial Statistics (CS1)

A random variable X has density f(x) = 3x^2 for 0 < x < 1 and zero elsewhere. What is P(X > 0.5)?

  1. 0.875 Correct answer.
  2. 0.125
  3. 0.250
  4. 0.500

Explanation

Integrating the density gives P(X > 0.5) = ∫_{0.5}^{1} 3x^2 dx = [x^3]_{0.5}^{1} = 1 − 0.125 = 0.875. The density integrates to one over (0,1).

Question 14Actuarial Statistics (CS1)

If X ~ Gamma(α, λ) with density proportional to x^{α−1} e^{−λx}, what are the mean and variance of X?

  1. α/λ and α/λ^2 Correct answer.
  2. α/λ and α^2/λ
  3. λ/α and λ/α^2
  4. αλ and αλ^2

Explanation

For this parameterisation the gamma mean is α/λ and the variance is α/λ^2. These follow from the moment generating function or standard results for the gamma family.

Question 15Actuarial Statistics (CS1)

In a simple linear regression with independent errors of constant variance, which property holds for the ordinary least squares slope estimator?

  1. It is biased but consistent
  2. It minimises the sum of absolute residuals
  3. It is the best linear unbiased estimator under the Gauss-Markov assumptions Correct answer.
  4. It always has a normal distribution regardless of error distribution

Explanation

The Gauss-Markov theorem states that OLS is the best linear unbiased estimator when errors have zero mean, equal variance, and are uncorrelated. Normality is only needed for exact inference, not for unbiasedness or efficiency among linear estimators.

Question 16Actuarial Statistics (CS1)

A risk X has a Pareto distribution with survival function S(x) = (θ/(x+θ))^α for x > 0. Which statement about its tail is correct?

  1. All moments of X are finite
  2. The mean is finite only when α > 1 Correct answer.
  3. The variance is finite for all α > 0
  4. The distribution is light-tailed with exponential decay

Explanation

For the Pareto distribution, E[X] exists only when α > 1, and higher moments require α to exceed the moment order. The tail decays polynomially, making it heavy-tailed.

Question 17Actuarial Statistics (CS1)

In a Markov chain, a state i is recurrent if which condition holds?

  1. The chain spends zero long-run proportion of time in i
  2. The probability of ever returning to i starting from i equals 1 Correct answer.
  3. The expected return time to i is finite
  4. State i can reach every other state

Explanation

Recurrence means return to the state is certain, so the probability of eventual return is one. Positive recurrence additionally requires a finite mean return time.

Question 18Actuarial Statistics (CS1)

Which statement correctly distinguishes the collective risk model from the individual risk model?

  1. The collective model fixes the number of policies while the individual model randomises it
  2. The individual model requires claim counts to be Poisson
  3. The collective model treats the number of claims as random, while the individual model sums fixed per-policy claim variables Correct answer.
  4. The collective model cannot accommodate a compound Poisson structure

Explanation

In the collective risk model the aggregate claim amount is a random sum with a random claim count, typically compound Poisson. The individual model sums a fixed number of policy-level claim variables.

Question 19Actuarial Statistics (CS1)

An insurer applies a deductible d to a loss X. For a loss exceeding d, the insurer pays X − d. Which expression gives the expected payment per loss E[(X − d)^+]?

  1. E[X] + d
  2. d·P(X > d)
  3. E[X] − d
  4. E[X] − d·P(X > d) Correct answer.

Explanation

The expected payment with a deductible is E[(X − d)^+] = ∫_d^∞ S(x) dx, which equals E[X] − E[min(X, d)]. Equivalently it can be written E[X] − d·P(X > d) when the lower part is accounted for correctly under the standard identity.

Question 20Actuarial Statistics (CS1)

A survival model assumes a constant force of mortality μ = 0.02 per year. What is the expected future lifetime of a life aged 50?

  1. 20 years
  2. 100 years
  3. 50 years Correct answer.
  4. 0.02 years

Explanation

With a constant force of mortality the future lifetime is exponential with mean 1/μ. Here 1/0.02 = 50 years, independent of the current age.

Question 21Actuarial Statistics (CS1)

In a credibility model, the credibility factor Z for a group with n observations and between-group variance structure is often Z = n/(n + k). What happens to Z as n increases?

  1. Z approaches 0, giving more weight to the prior mean
  2. Z stays constant regardless of n
  3. Z becomes negative for large n
  4. Z approaches 1, giving more weight to the group's own experience Correct answer.

Explanation

As the group's own data grows, credibility increases toward full weight on the observed experience. The factor Z = n/(n+k) increases monotonically toward 1 as n grows.

Question 22Actuarial Statistics (CS1)

Which distribution is the canonical choice for modelling the number of claims when events occur independently at a constant rate over time?

  1. Binomial distribution
  2. Normal distribution
  3. Poisson distribution Correct answer.
  4. Lognormal distribution

Explanation

Independent events at a constant rate produce a Poisson count, whose mean and variance both equal the rate times the exposure. The binomial requires a fixed maximum number of trials.

Question 23Actuarial Statistics (CS1)

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

  1. It models the logarithm of the mean response Correct answer.
  2. It models the mean response directly on the identity scale
  3. It models the reciprocal of the mean response
  4. It models the variance of the response

Explanation

A log link sets the linear predictor equal to log of the mean, so the mean is the exponential of the linear predictor. This ensures fitted means are positive.

Question 24Actuarial Statistics (CS1)

In a two-state Markov model for disability (Healthy and Disabled) with transition intensities, which quantity represents the force of transition from Healthy to Disabled?

  1. The probability of being Disabled at age 65
  2. The instantaneous rate of transition from Healthy to Disabled at a given age Correct answer.
  3. The proportion of lives Disabled at the start
  4. The expected time spent Disabled over a lifetime

Explanation

Transition intensities are instantaneous rates governing movement between states in continuous time. They are not probabilities of being in a state at a fixed age.

Question 25Actuarial Statistics (CS1)

A portfolio has independent annual claim counts with mean 100 and variance 150. Using the normal approximation, which expression gives an approximate 99% upper bound on the total count?

  1. 100 − 2.326·√150
  2. 100 + 2.326·√150 Correct answer.
  3. 100 + 1.645·√150
  4. 100 + 2.326·150

Explanation

The normal approximation uses mean plus the z-value times the standard deviation. For a 99% one-sided bound the z-value is about 2.326 and the standard deviation is √150.

Question 26Actuarial Statistics (CS1)

A life table gives l_40 = 95,000 and l_41 = 94,525. Using the standard approximation for the probability that a life aged 40 dies within the next year, q_40, what is its value?

  1. 0.00525
  2. 0.9950
  3. 0.0050 Correct answer.
  4. 0.00475

Explanation

Under the life table, q_40 = (l_40 − l_41)/l_40 = (95,000 − 94,525)/95,000 = 475/95,000 = 0.005. This is the standard one-year death probability for a life aged exactly 40.

Question 27Actuarial Statistics (CS1)

In a survival analysis, a censored observation at time t contributes which term to the likelihood for a parametric survival model with density f and survival function S?

  1. 1 − S(t)
  2. f(t)
  3. f(t)/S(t)
  4. S(t) Correct answer.

Explanation

A right-censored observation tells us only that the survival time exceeds t, so its likelihood contribution is the probability of surviving to t, namely S(t). Uncensored observations contribute the density f(t).

Question 28Actuarial Statistics (CS1)

A compound Poisson model has claim count N ~ Poisson(λ) and individual claim amounts with mean μ and variance σ², independent of N. Which expression gives the variance of the aggregate claim S = X_1 + ... + X_N?

  1. λσ²
  2. λ²σ²
  3. λμ
  4. λσ² + λμ² Correct answer.

Explanation

By the conditional variance formula, Var(S) = E[N]Var(X) + Var(N)(E[X])² = λσ² + λμ². This combines the variability of individual claim sizes with the variability of the count.

Question 29Actuarial Statistics (CS1)

For a compound Poisson aggregate claims distribution with Poisson parameter λ and claim amount distribution F, which statement about the Panjer recursion is correct?

  1. It computes the aggregate distribution recursively from the claim size probabilities using the relation for the Poisson class. Correct answer.
  2. It applies only to the individual risk model with a fixed number of policies.
  3. It gives the exact distribution only when all claims are of equal size.
  4. It requires the claim sizes to be normally distributed.

Explanation

The Panjer recursion exploits the (a, b, 0) class structure of the Poisson counting distribution to compute aggregate probabilities from the discrete claim size distribution. It is exact for discrete claim amounts and does not require normal claims or a fixed portfolio.

Question 30Actuarial Statistics (CS1)

An insurer uses proportional reinsurance with retention α (0 < α < 1). For a ground-up loss X, which expression gives the reinsurer's payment?

  1. (1 − α)X Correct answer.
  2. min(X, α)
  3. αX
  4. max(0, X − α)

Explanation

Under proportional reinsurance the cedant retains a fraction α of each loss, so the reinsurer pays the remaining fraction (1 − α)X. The split is proportional to the loss for every loss size.

Question 31Actuarial Statistics (CS1)

A surplus (or collective risk) process has premium income c per unit time and aggregate claims modelled by a compound Poisson process with Poisson rate λ and claim size distribution F. Which condition is required for the adjustment coefficient R to exist as a positive root?

  1. The premium loading is positive, so that c exceeds the expected claims per unit time. Correct answer.
  2. The claim size distribution must have bounded support.
  3. The Poisson rate λ must exceed one.
  4. The claim sizes must be exponentially distributed.

Explanation

A positive safety loading ensures the premium income exceeds expected claims, which guarantees a positive solution R to the Lundberg equation. This is the condition for the adjustment coefficient and hence for exponential decay of ruin probability.

Question 32Actuarial Statistics (CS1)

In a Markov jump process used to model a life with states Healthy and Dead, the transition intensity from Healthy to Dead is μ. Starting in Healthy, what is the probability of remaining Healthy for at least t years?

  1. μt
  2. 1 − exp(−μt)
  3. exp(−μt) Correct answer.
  4. 1/(1 + μt)

Explanation

With a single constant transition intensity μ out of the Healthy state, the waiting time in Healthy is exponential with rate μ. The survival probability is therefore exp(−μt).

Question 33Actuarial Statistics (CS1)

A survival model uses the Makeham law with force of mortality μ(x) = A + Bc^x. Which statement about the survival function is correct?

  1. It is a linear function of x.
  2. It is exp(−Ax − (B/ln c)(c^x − 1)) for the appropriate constants. Correct answer.
  3. It is exp(−Bc^x).
  4. It is exp(−Ax) only, since the Makeham term integrates to zero.

Explanation

Integrating μ from age 0 to x gives Ax + (B/ln c)(c^x − 1), and the survival function is the exponential of the negative integral. This is the standard Makeham survival function.

Question 34Actuarial Statistics (CS1)

A random variable has moment generating function M(t) = (1 − t/λ)^{−α} for t < λ. Which distribution does this correspond to?

  1. Gamma with shape α and rate λ Correct answer.
  2. Poisson with mean α/λ
  3. Exponential with rate λ
  4. Normal with mean α and variance λ

Explanation

The mgf (1 − t/λ)^{−α} is the standard gamma mgf with shape α and rate λ. Its mean is α/λ and variance α/λ².

Question 35Actuarial Statistics (CS1)

If X ~ Exponential(λ) and Y = min(X, m) for a fixed m > 0, what is E[Y]?

  1. m
  2. 1/λ
  3. exp(−λm)/λ
  4. (1 − exp(−λm))/λ Correct answer.

Explanation

E[min(X,m)] = ∫₀^m S(x) dx for a positive random variable, and S(x) = exp(−λx). Integrating gives (1 − exp(−λm))/λ, which is bounded above by 1/λ.

Question 36Actuarial Statistics (CS1)

In a credibility setting, the Bühlmann credibility factor for a risk with n observations is Z = n/(n + k). What does the parameter k represent?

  1. The ratio of the expected process variance to the variance of the hypothetical means. Correct answer.
  2. The overall portfolio mean.
  3. The number of observations available.
  4. The variance of the observed sample.

Explanation

In Bühlmann credibility, k = E[σ²(θ)]/Var(μ(θ)), the ratio of within-risk (process) variance to between-risk variance of the hypothetical means. A larger k means less credibility is given to the individual experience.

Question 37Actuarial Statistics (CS1)

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

  1. The link function must be the identity for the Poisson family.
  2. The expected response is constrained to be positive and is modelled as exp(β^T x). Correct answer.
  3. The response is assumed to be normally distributed with constant variance.
  4. The linear predictor must be positive.

Explanation

With a log link, the mean is μ = exp(β^T x), which is automatically positive for any real linear predictor. The linear predictor itself is unrestricted in sign.

Question 38Actuarial Statistics (CS1)

In Bayesian estimation, a prior distribution is described as conjugate for a likelihood if which property holds?

  1. The posterior is always normal.
  2. The prior mean equals the maximum likelihood estimate.
  3. The prior has a finite variance.
  4. The posterior distribution belongs to the same family as the prior. Correct answer.

Explanation

Conjugacy means the posterior stays within the same parametric family as the prior, which makes updating analytically tractable. For example, a beta prior with a binomial likelihood yields a beta posterior.

Question 39Actuarial Statistics (CS1)

A sample of n independent observations is drawn from N(μ, σ²) with both parameters unknown. Which pivot is used to construct a confidence interval for μ?

  1. (X̄ − μ)/(σ/√n), which follows a standard normal distribution
  2. (n − 1)S²/σ², which follows a chi-square distribution
  3. (X̄ − μ)/(S/√n), which follows a t distribution with n − 1 degrees of freedom Correct answer.
  4. (X̄ − μ)/S, which follows a t distribution with n degrees of freedom

Explanation

When σ is unknown and estimated by the sample standard deviation S, the pivot (X̄ − μ)/(S/√n) follows a t distribution with n − 1 degrees of freedom. This yields the standard t-based confidence interval for the mean.

Question 40Actuarial Statistics (CS1)

A hypothesis test rejects the null hypothesis when the p-value is below the significance level α. Which statement about the relationship between the p-value and the test statistic is correct?

  1. The p-value is the probability of rejecting a true null hypothesis.
  2. The p-value is the probability that the null hypothesis is true given the data.
  3. The p-value is the probability, computed under the null hypothesis, of a test statistic at least as extreme as the observed one. Correct answer.
  4. The p-value is the probability that the alternative hypothesis is true.

Explanation

The p-value is a tail probability of the test statistic calculated assuming the null hypothesis holds, measuring how extreme the observed result is. It is not the probability that either hypothesis is true.

Question 41Actuarial Statistics (CS1)

Two independent samples give X̄₁ = 10 with n₁ = 50 and X̄₂ = 12 with n₂ = 50, from populations with common known variance σ² = 100. What is the standard error of the difference X̄₁ − X̄₂?

  1. √2
  2. 1
  3. 4
  4. 2 Correct answer.

Explanation

The variance of the difference is σ²/n₁ + σ²/n₂ = 100/50 + 100/50 = 4, so the standard error is √4 = 2. Known variances and independence are used directly.

Question 42Actuarial Statistics (CS1)

In a simple linear regression, the residual sum of squares is minimised by the ordinary least squares estimates. Which statement about the fitted line is correct?

  1. It maximises the sum of squared residuals.
  2. It passes through the origin whenever the intercept is estimated.
  3. It passes through the point (X̄, Ȳ). Correct answer.
  4. It has slope equal to the ratio of the sample variances.

Explanation

The normal equations imply that the OLS line with an intercept always passes through the sample means (X̄, Ȳ). This is a direct consequence of the residuals summing to zero.

Question 43Actuarial Statistics (CS1)

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

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

Explanation

E[X] = ∫₀¹ x·2x dx = ∫₀¹ 2x² dx = 2/3. The density integrates to one and gives a mean greater than one-half because it is skewed toward one.

Question 44Actuarial Statistics (CS1)

For a discrete-time Markov chain with transition matrix P, a stationary distribution π satisfies which condition?

  1. πP = 0
  2. P² = P
  3. Pπ = π with π summing to one
  4. πP = π with the components of π summing to one Correct answer.

Explanation

A stationary distribution is a probability vector left-invariant under the transition matrix, so πP = π and the components sum to one. It describes the long-run proportion of time spent in each state under appropriate conditions.

Question 45Actuarial Statistics (CS1)

A Poisson process has rate λ = 3 events per hour. Given that exactly one event occurred in a one-hour interval, what is the probability it occurred in the first 20 minutes?

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

Explanation

Conditional on exactly one event in [0,1], its time is uniformly distributed on that interval. The probability it falls in the first 20 minutes (one third of the hour) is therefore 1/3.

Question 46Actuarial Statistics (CS1)

In a life insurance context, the present value random variable for a whole life insurance paying 1 at the moment of death is v^T where T is the future lifetime. Which statement about its expected value is correct?

  1. It equals the integral over t of v^t times the density of T. Correct answer.
  2. It equals the force of interest times E[T].
  3. It equals v^{E[T]}.
  4. It is always greater than one.

Explanation

The actuarial present value is E[v^T] = ∫ v^t f_T(t) dt, an expectation over the distribution of the future lifetime. Because v^t is convex, Jensen's inequality shows it exceeds v^{E[T]}.

Question 47Actuarial Statistics (CS1)

An annuity-due of 1 per year is payable annually to a life aged x for as long as the life survives, with at most n payments. Which expression represents its actuarial present value?

  1. The sum over k = 1 to n of v^k times the probability of death in year k
  2. The sum over k = 0 to n−1 of v^k times the probability of survival to age x+k Correct answer.
  3. The expected future lifetime discounted at rate i
  4. n times v^n

Explanation

An annuity-due pays at the start of each year, so the payment at time k (k = 0,...,n−1) occurs with probability ₖp_x of survival. Discounting each expected payment at v^k and summing gives the actuarial present value.

Question 48Actuarial Statistics (CS1)

A survival distribution has hazard function h(t) = 1/(t + 1) for t ≥ 0. Which statement about the corresponding survival function S(t) is correct?

  1. S(t) = (t + 1), which is not a valid survival function.
  2. S(t) = exp(−t), so the distribution is exponential.
  3. S(t) = 1/(t + 1), so the distribution is heavy-tailed. Correct answer.
  4. S(t) = 1 − 1/(t + 1), so the density is bounded.

Explanation

Integrating the hazard gives H(t) = ln(t + 1), so S(t) = exp(−H(t)) = 1/(t + 1). This is a Pareto-type survival function whose tail decays polynomially, hence heavy-tailed.

Question 49Actuarial Statistics (CS1)

In the individual risk model, the aggregate claim is the sum of independent individual claim variables S_i, each zero with high probability. Which statement about the variance of the aggregate claim is correct?

  1. It equals the sum of the individual variances, since the S_i are assumed independent. Correct answer.
  2. It is always larger than the variance under the collective risk model.
  3. It cannot be computed without assuming Poisson claim counts.
  4. It equals the square of the sum of the individual means.

Explanation

Under independence, Var(Σ S_i) = Σ Var(S_i), and each Var(S_i) = q_i(1 − q_i)μ_i² + q_iσ_i² for claim probability q_i. No Poisson assumption is needed in the individual model.

Question 50Actuarial Statistics (CS1)

A statistician compares two nested models using a likelihood ratio test. Under the null hypothesis that the smaller model is adequate, which distribution does twice the log-likelihood difference approximately follow?

  1. A chi-square distribution with degrees of freedom equal to the number of extra parameters Correct answer.
  2. An F distribution with one and n degrees of freedom
  3. A t distribution with n − 1 degrees of freedom
  4. A normal distribution with mean zero and unit variance

Explanation

Wilks' theorem states that 2(log L_full − log L_reduced) is asymptotically chi-square, with degrees of freedom equal to the difference in the number of parameters. This underpins the likelihood ratio test.