CS1 Actuarial Statistics
DIFFERENTIATE BETWEEN CONTINUOUS RANDOM VARIABLE & DISCRETE RANDOM VARIABLE?
Discrete random variables are random variables that take a finite number of values. For example, the outcome of rolling a die. Continuous random variables, on the other hand, can take on any value in a given interval.
WHAT IS LAW OF LARGE NUMBERS?
The Law of Large Numbers states that as the number of observations increases, the sample average converges to the expected value of the population. It helps in predicting average outcomes more accurately over a large number of trials. It basically helps in predicting the losses in a better manner.
WHAT DO YOU MEAN BY POISSON PROCESS?
A Poisson process models the number of events occurring in a fixed interval of time or space, where the events happen independently and at a constant average rate.
HOW MONTE CARLO SIMULATION TAKES PLACE?
A Monte Carlo simulation is used to predict the probability of a variety of outcomes when there are random variables present. Monte Carlo simulations help to explain the impact of risk and uncertainty in prediction and forecasting models.
WHAT IS THE SIGNIFICANCE OF P-VALUE?
The p-value, also called probability value, is the lowest level at which H0 (null hypothesis) can be rejected.
GIVE AN EXAMPLE OF APPLICATION OF BINOMIAL DISTRIBUTION.
Examples of binomial distribution include the number of heads in a fixed number of coin tosses, number of defective items in a batch, or number of students passing an exam (success/failure situations).
WHAT IS CENTRAL LIMIT THEOREM?
Central Limit Theorem gives us an approximate distribution of the sample mean when the population distribution is unknown and more importantly does not need to be known. It provides useful normal approximations to the distributions of particular functions.
WHAT DOES LEVEL OF SIGNIFICANCE DEPICT?
It is the probability of rejecting H0 when it is in fact true.
WHAT IS POSTERIOR DISTRIBUTION? HOW IS IT RELATED TO PRIOR DISTRIBUTION?
The conditional distribution given the observed data is called the posterior distribution of theta. If the prior distribution is continuous, then the posterior distribution is also continuous. Similarly, if the prior distribution is discrete, then the posterior distribution is also discrete.
WHAT IS CREDIBILITY FACTOR?
The credibility premium formula for this risk is Z*X_BAR+(1-Z)*MU where Z is a number between zero and one and is known as the credibility factor.
WHAT IS MEANT BY CAUSATION EFFECT?
Causation refers to a relationship where a change in one variable directly causes a change in another. Spurious correlation means two variables appear related but have no causal connection.
WHAT IS SENSITIVITY AND SPECIFICITY?
Sensitivity refers to the true positive rate whereas specificity refers to the true negative rate. For example,the ability of a test to correctly identify patients with a disease is sensitivity whereas specificity is the ability of a test to correctly identify people without the disease
WHAT IS R SQUARED AND ADJUSTED R SQUARE? HOW ARE THEY RELATED?
the proportion of the total variation of the responses ‘explained’ by a model, called the coefficient of determination, denoted R-square whereas adjusted r-square gives a measure of how much variability is explained by the regression model. It takes account of the undesirability of increased complexity by the r-square method.
NAME SOME DISTRIBUTIONS BELONGING TO EXPONENTIAL FAMILY.
Normal, poisson, binomial, gamma, lognormal.
DIFFERENCE BETWEEN SAMPLE VARIANCE AND POPULATION VARIANCE.
Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. Sample variance is an unbiased estimator of the population variance.
WHAT IS PRINCIPAL COMPONENT ANALYSIS? WHAT IS ITS OBJECTIVE?
Principal Component Analysis (PCA) is a technique used to reduce the dimensionality of large data sets by transforming variables into a smaller number of uncorrelated components that retain most of the original variance.
WHAT IS A SATURATED MODEL?
A saturated model is defined to be a model in which there are as many parameters as observations, so that the fitted values are equal to the observed values.
WHAT DOES CONTINGENCY TABLE SIGNIFY?
A contingency table consists of rows and columns containing counts of sample items (people, claims etc) that are classified according to two category variables.
WHAT IS CONDITIONAL EXPECTATION?
The conditional expectation of Y given X =x is the mean of the conditional distribution of Y given X= x.
DIFFERENCE BETWEEN COVARIANCE AND CORRELATION.
Covariance refers to the relationship between two random variables in which a change in the other reflects a change in one variable which can range from -∞ to +∞. Correlation determines the degree to which two or more random variables move in sequence. Its value ranges from -1 to 1.
What is Memoryless Property?
The Memoryless Property means that the future probability of an event is completely independent of the past.This property is unique to only two distributions: the Exponential distribution (continuous) and the Geometric distribution (discrete).
Why is joint distribution important in risk or financial analysis?
Joint distribution is important because financial and risk variables rarely behave independently. It helps capture how multiple variables move together, which is critical when analysing combined outcomes such as total risk, portfolio returns, or aggregated losses. Ignoring joint behaviour can lead to underestimating or misjudging overall risk.
What is the difference between marginal and joint distributions?
A marginal distribution describes the behaviour of a single variable on its own, while a joint distribution describes the combined behaviour of two or more variables together. Marginal distributions ignore dependence, whereas joint distributions explicitly account for relationships between variables.
Are two variables independent if they are uncorrelated?
No. Uncorrelated variables have zero linear relationship, but they may still be dependent in a non-linear way. Independence is a much stronger condition and requires that the joint distribution can be expressed as the product of the marginal distributions.
What is conditional expectation?
Conditional expectation represents the expected value of a variable given that certain information is already known. It reflects how expectations are updated when new information becomes available.
Why is conditional expectation important in financial or risk analysis?
Conditional expectation is important because most financial decisions are made with partial information. It allows forecasts, valuations, and risk estimates to be adjusted dynamically as new data or signals are observed.
What does the Central Limit Theorem states?
The Central Limit Theorem states that, under fairly general conditions, the distribution of the sample mean tends to become approximately normal as the sample size increases, regardless of the underlying distribution of the data.
Why is the Central Limit Theorem important in data analysis or finance?
The Central Limit Theorem allows normal-based methods to be used even when the underlying data is not normally distributed. This enables estimation, hypothesis testing, confidence intervals, and risk assessment using relatively simple and well-understood tools.
Does the Central Limit Theorem require the original data to be normally distributed?
No. The theorem does not require the original data to be normally distributed. It only requires that observations are independent and have finite mean and variance, with normality emerging as the sample size increases.
Can the Central Limit Theorem always be applied in practice?
No. The theorem may not work well when sample sizes are very small, data is highly skewed, observations are dependent, or the underlying distribution has infinite variance. In such cases, normal approximations can be misleading.
What is sampling and why is it used instead of population data?
Sampling involves selecting a subset of data from a larger population to draw conclusions about the population as a whole. It is used because collecting full population data is often costly, time-consuming, or impractical, while a well-designed sample can provide reliable insights.
What is sampling bias and why is it a problem?
Sampling bias occurs when certain segments of the population are over-represented or under-represented in the sample. This leads to distorted estimates and incorrect inferences about the population.
What is statistical inference?
Statistical inference is the process of using sample data to make conclusions or decisions about a population while explicitly accounting for uncertainty.
Why is sample size important in inference?
Sample size affects the reliability of inference. Larger samples generally lead to more precise estimates and narrower confidence intervals, while small samples increase uncertainty and reduce statistical power.
What assumptions underlie most statistical inference methods?
Most inference methods assume random sampling, independence of observations, and a reasonable approximation of the underlying distribution. Violations of these assumptions can weaken the validity of conclusions.
What is the chi-square test typically used for?
The chi-square test is used to assess whether there is a significant difference between observed and expected frequencies, often to test independence or goodness of fit.
What assumptions underlie the chi-square test?
Key assumptions include independence of observations and sufficiently large expected frequencies to ensure the approximation is valid.
When is a t-test preferred over a normal or z-test?
A t-test is preferred when sample sizes are small and the population variance is unknown, as it accounts for additional uncertainty in estimating variability.
What does a t-test actually test?
A t-test evaluates whether the difference between sample means is statistically significant relative to the variability observed in the data.
What is the purpose of an F-test?
An F-test is primarily used to compare variances or to assess whether a group of variables collectively has explanatory power in a model.
Where is the F-test commonly encountered in practice?
The F-test commonly appears in regression analysis, model comparison, and variance analysis to test overall model significance or equality of variances.
What is meant by normal approximation and when is it used?
Normal approximation refers to using the normal distribution to approximate the distribution of a statistic when the sample size is sufficiently large. It is commonly used to simplify inference when exact distributions are difficult to work with.
What is the difference between a population-based test and a sample-based test?
Population-based tests assume that the population parameters or distribution are known or specified, allowing direct probability calculations. Sample-based tests rely on sample data to estimate unknown population parameters and make inferences while accounting for sampling uncertainty.
When is a binomial test used instead of a normal test?
A binomial test is used when outcomes are discrete, such as success or failure, and when the sample size is small or an exact test is required. A normal test is preferred when the sample size is large and a normal approximation is appropriate.
What is the Method of Moments?
The Method of Moments estimates population parameters by equating theoretical moments of a distribution to corresponding sample moments.
When is Method of Moments preferred?
It is preferred when simplicity and computational ease are important, especially when likelihood functions are complex or difficult to maximise.
What are the limitations of Method of Moments?
Method of Moments estimators may not be efficient, can be biased, and do not always make full use of the information available in the data.
What is the idea behind Maximum Likelihood Estimation?
Maximum Likelihood Estimation chooses parameter values that maximise the probability of observing the given sample data under the assumed model.
Is an MLE always unbiased?
No. MLEs can be biased in finite samples, although they are often consistent and asymptotically unbiased.
How does MLE compare to Method of Moments?
MLE generally produces more efficient estimators than Method of Moments but may be computationally more complex.
What are the key desirable properties of an estimator?
Key properties include unbiasedness, consistency, efficiency, and low variance.
What is the Cramér–Rao Lower Bound?
The Cramér–Rao Lower Bound provides a theoretical lower limit on the variance of any unbiased estimator of a parameter.
Why is CRLB important in estimation theory?
It serves as a benchmark to evaluate how efficient an estimator is compared to the best possible unbiased estimator.
What is a confidence interval?
A confidence interval provides a range of values that is likely to contain the true population parameter, based on sample data and a specified confidence level.
What is a prediction interval?
A prediction interval provides a range within which a future individual observation is likely to fall, taking into account both estimation uncertainty and inherent variability.
What is the key difference between a confidence interval and a prediction interval?
A confidence interval estimates an unknown population parameter, while a prediction interval estimates the range of a future observation. Prediction intervals are wider because they include additional uncertainty.
What is hypothesis testing?
Hypothesis testing is a statistical framework used to make decisions about population parameters based on sample data, while controlling the risk of incorrect conclusions.
What is a null hypothesis and why is it important?
The null hypothesis represents the default or status-quo assumption and serves as a benchmark against which evidence from the data is evaluated.
What is the alternative hypothesis?
The alternative hypothesis represents a competing claim that is supported when sufficient evidence exists against the null hypothesis.
What is a Type I error?
A Type I error occurs when a true null hypothesis is incorrectly rejected.
What is a Type II error?
A Type II error occurs when a false null hypothesis is not rejected.
What is the trade-off between Type I and Type II errors?
Reducing the probability of one type of error typically increases the probability of the other, requiring a balance based on context.
What is a significance level?
The significance level acts as a risk threshold; it is the maximum probability we are willing to accept of concluding an effect exists when it is actually just random chance (a Type I error).
What does a p-value represent?
The p-value measures the strength of evidence against the null hypothesis by quantifying how extreme the observed data is under the null assumption.
What is the difference between one-sided and two-sided tests?
One-sided tests detect effects in a specified direction, while two-sided tests detect effects in either direction.
What is a test statistic?
A test statistic is a numerical summary calculated from sample data that is used to assess evidence against the null hypothesis.
Why is a test statistic needed in hypothesis testing?
It converts sample data into a standardized measure that can be compared against a reference distribution to make a decision.
What is a critical value?
A critical value is a threshold determined by the chosen significance level, beyond which the null hypothesis is rejected.
What is correlation?
Correlation measures the strength and direction of the linear relationship between two variables.
What does a correlation of zero imply?
A correlation of zero implies no linear relationship, but it does not rule out non-linear dependence.
Does high correlation imply causation?
No. Correlation indicates association, not causation. Two variables may move together due to coincidence or a third underlying factor.
What is Pearson correlation used for?
Pearson correlation measures the strength and direction of a linear relationship between two continuous variables.
What is Spearman correlation?
Spearman correlation measures the strength and direction of a monotonic relationship using ranked data.
What is Kendall’s tau correlation?
Kendall’s tau measures association based on the concordance and discordance of ranked pairs.
How does Pearson differ from Spearman and Kendall?
Pearson measures linear relationships using raw values, while Spearman and Kendall measure monotonic relationships using ranks.
What is multivariate correlation analysis?
Multivariate correlation analysis examines the relationships among more than two variables simultaneously to understand overall dependence structure.
What is linear regression?
Linear regression models the relationship between a dependent variable and one or more independent variables by fitting a linear equation.
What does the regression coefficient represent?
A regression coefficient represents the expected change in the dependent variable for a one-unit change in the independent variable, holding other variables constant.
What is the interpretation of the intercept?
The intercept represents the expected value of the dependent variable when all independent variables are zero, though it may not always have practical meaning.
What does R-squared measure?
R-squared measures the proportion of variability in the dependent variable explained by the model.
What is adjusted R-squared?
Adjusted R-squared accounts for the number of predictors and penalises unnecessary complexity
What does the t-test on a regression coefficient test?
It tests whether a coefficient is significantly different from zero.
What is SST (Total Sum of Squares)?
SST measures the total variability in the dependent variable around its mean.
What is SSR (Regression Sum of Squares)?
SSR measures the portion of total variability that is explained by the regression model.
What does R² actually measure?
R² measures goodness of fit, not predictive accuracy or causality.
What is multiple linear regression?
Multiple linear regression models the relationship between a dependent variable and multiple independent variables simultaneously.
What is a Generalised Linear Model (GLM)?
A Generalised Linear Model extends linear regression to handle non-normal response variables by linking the mean of the response to predictors through a suitable link function.
Why was GLM introduced when linear regression already exists?
Linear regression assumes normally distributed errors and constant variance, which is unsuitable for many real-world outcomes such as counts, proportions, or binary data. GLMs address this limitation.
What are the three main components of a GLM?
A random component specifying the distribution of the response, a systematic component formed by predictors, and a link function connecting the two.
What is the role of the link function in a GLM?
The link function connects the expected value of the response variable to a linear combination of predictors.
Why is the choice of link function important?
An inappropriate link function can lead to poor model fit, biased interpretation, or predictions outside the valid range.
When is a Poisson GLM used?
When modelling count data such as number of events occurring in a fixed period.
When is a binomial GLM used?
When modelling binary outcomes or proportions.
Why is the log link commonly used with Poisson models?
It ensures predicted values remain positive and allows multiplicative interpretation.
Why is the logit link used with binomial models?
It maps values to a bounded range and provides interpretable odds-based effects.
How are coefficients interpreted in a GLM?
Coefficients represent the effect of predictors on the transformed mean of the response, as defined by the link function.
Why do GLM coefficients not represent simple additive changes?
Because the relationship between predictors and the response is mediated through a non-linear link function.
How does interpretation differ between identity and log links?
Identity links imply additive effects, while log links imply multiplicative effects.
How is goodness of fit assessed in GLMs?
Through residual analysis, deviance measures, and comparison of nested models.
What are deviance residuals?
They measure the contribution of each observation to the overall model deviance.
What is Bayesian statistics?
Bayesian statistics is an approach to inference where prior beliefs are updated using observed data to obtain posterior beliefs.
How does Bayesian inference differ from classical (frequentist) inference?
Bayesian inference treats parameters as uncertain quantities and updates beliefs with data, while classical inference treats parameters as fixed and focuses on long-run sample behaviour.
What is meant by prior information in Bayesian analysis?
Prior information represents existing knowledge or assumptions about a parameter before observing current data.
What is a posterior distribution?
The posterior distribution represents updated beliefs about a parameter after combining prior information with observed data.
What is the difference between informative and non-informative priors?
Informative priors reflect strong prior knowledge, while non-informative priors aim to minimise prior influence.
When would an informative prior be preferred?
When reliable historical data or expert knowledge is available.
What are the risks of using strong priors?
They can dominate the data and lead to biased conclusions if the prior assumptions are incorrect.
How are confidence intervals different from Bayesian credible intervals?
Credible intervals represent ranges where the parameter is believed to lie given the data, while confidence intervals describe properties of repeated sampling.
Why is Bayesian analysis popular in risk and forecasting problems?
Because it naturally incorporates uncertainty and updates predictions as new information becomes available.”
What is credibility theory?
Credibility theory is a statistical framework used to combine individual experience with collective experience to produce more reliable estimates.
Why is credibility theory needed?
Because individual data may be limited or volatile, while collective data may be stable but less specific. Credibility theory balances both.
What is meant by direct data in credibility theory?
Direct data refers to experience that comes from the specific individual or risk being analysed.
What is collateral data?
Collateral data refers to experience from similar individuals or the wider group that provides additional context.
What is a credibility premium?
A credibility premium is a weighted estimate formed by combining individual experience with collective experience.
What is the credibility factor?
The credibility factor represents the weight assigned to individual experience in forming the credibility premium.
When is the Poisson–Gamma model used?
It is used when modelling count-based outcomes with variability across individuals.
When is the Normal–Normal model used?
It is used when outcomes are continuous and variability can be reasonably approximated by normal distributions.
Why are Poisson–Gamma and Normal–Normal models popular in credibility theory?
Because they provide mathematically consistent and interpretable credibility structures.
What is Empirical Bayes credibility?
Empirical Bayes credibility is an approach where credibility weights are derived by estimating prior parameters from the observed data itself rather than assuming them to be known.
What is EBCT-1 in credibility theory?
EBCT-1 is an empirical Bayesian approach where the credibility premium is obtained by estimating structural parameters from observed data under simplifying assumptions.
What is EBCT-2 in credibility theory?
EBCT-2 is an extension of EBCT-1 that relaxes simplifying assumptions and allows greater flexibility in modelling variability.
Why is EBCT-2 considered more robust than EBCT-1?
Because it relies less on restrictive assumptions and better reflects portfolio diversity.
