Question 1

How does the Bernoulli distribution relate to the Binomial, and what are their parameters and moments?

Accepted Answer

A Bernoulli(p) trial is the atomic unit: a single experiment with success probability p. Binomial(n, p) is the sum of n independent, identically distributed Bernoulli(p) trials, counting total successes. Because Binomial is a sum of independent random variables, its mean and variance are n times those of a single Bernoulli.

Question 2

What is the chi-square test, and when do you use it?

Accepted Answer

The chi-square test assesses whether observed categorical frequencies differ from expected frequencies (goodness-of-fit) or whether two categorical variables are independent of each other (test of independence). It requires count data, a sufficiently large sample, and expected cell counts of at least 5.

Question 3

What is the difference between permutations and combinations, and when does each apply?

Accepted Answer

Permutations count ordered arrangements: P(n,k) = n!/(n-k)!. Combinations count unordered selections: C(n,k) = n!/[k!(n-k)!]. The rule is simple — if the order of selection matters, permute; if it doesn't, combine. Combinations are always smaller by a factor of k!.

Question 4

What is conditional probability, and how does it differ from joint probability?

Accepted Answer

Conditional probability P(A|B) is the probability of A given that B has already occurred, computed as P(A and B) / P(B). It narrows the sample space to B, whereas joint probability P(A and B) lives in the full, unrestricted space.

Question 5

What is the difference between correlation and causation, and why does the distinction matter?

Accepted Answer

Correlation measures the strength of a linear relationship between two variables, but a shared cause, reverse causation, or coincidence can all produce correlation without any causal link. Treating correlation as causation leads to interventions that fail or cause harm.

Question 6

What is the difference between covariance and correlation, and when does each matter?

Accepted Answer

Covariance measures the direction of the linear relationship between two variables and is expressed in the product of their units, making it scale-dependent and hard to interpret across different variable pairs. Correlation normalises covariance by both standard deviations to produce a dimensionless measure bounded between -1 and 1, enabling comparison across pairs.

Question 7

Define expected value and variance. What are their key properties?

Accepted Answer

Expected value is the probability-weighted average outcome of a random variable; variance measures average squared deviation from that mean. Both are linear/additive in specific ways — knowing these rules prevents algebraic mistakes under interview pressure.

Question 8

What is the difference between independent events and mutually exclusive events?

Accepted Answer

Mutually exclusive events cannot both occur at once — knowing one happened tells you the other didn't. Independent events can both occur, but knowing one happened gives no information about the other. These concepts are nearly opposite: non-trivial mutually exclusive events are always dependent.

Question 9

State the law of total probability and give a concrete example of when you'd apply it.

Accepted Answer

The law of total probability decomposes P(A) over a mutually exclusive, exhaustive partition of the sample space: P(A) = Σ P(A|Bᵢ)·P(Bᵢ). It is the engine behind the Bayes denominator and any calculation where you want an overall rate built from segment-level rates.

Question 10

When should you use mean vs median vs mode, and which is most robust to outliers?

Accepted Answer

Mean is optimal for symmetric, outlier-free data; median is the go-to for skewed distributions or when outliers are real rather than errors; mode is the only sensible average for nominal/categorical data. Robustness is a formal concept — the median's breakdown point is 50%, meaning half the data can be corrupted before it fails, while the mean's breakdown point is essentially 0%.

Question 11

Explain the normal distribution and the 68-95-99.7 empirical rule.

Accepted Answer

The normal distribution is a symmetric, bell-shaped probability distribution completely described by its mean and standard deviation. The empirical rule states that approximately 68%, 95%, and 99.7% of observations fall within one, two, and three standard deviations of the mean respectively — a direct consequence of integrating the Gaussian density over those intervals.

Question 12

What is the difference between the null and alternative hypothesis?

Accepted Answer

The null hypothesis (H0) is the default claim of no effect or no difference, while the alternative hypothesis (H1) is what you are trying to find evidence for. Hypothesis testing asks whether the observed data is surprising enough under H0 to justify rejecting it in favor of H1.

Question 13

What is the difference between one-tailed and two-tailed hypothesis tests, and when is each appropriate?

Accepted Answer

A two-tailed test rejects H0 when the statistic is extreme in either direction; a one-tailed test rejects only in one pre-specified direction. Two-tailed tests are the default because they guard against effects in both directions; one-tailed tests are valid only when a directional hypothesis is theoretically justified and pre-registered before seeing the data.

Question 14

What is the significance level (alpha) in hypothesis testing, and how do you choose it?

Accepted Answer

The significance level alpha is the maximum tolerable probability of a Type I error — rejecting a true null hypothesis. It must be chosen before data collection based on the relative costs of false positives versus false negatives, not defaulted to 0.05 out of convention.

Question 15

What is the difference between standard error and standard deviation?

Accepted Answer

Standard deviation measures the spread of individual observations around the population mean. Standard error measures the spread of sample means around the true mean — it equals the standard deviation divided by the square root of the sample size, so it shrinks as the sample grows while the standard deviation does not.

Question 16

What is survivorship bias, and how does it distort statistical conclusions?

Accepted Answer

Survivorship bias occurs when analysis is restricted to the subset of observations that 'survived' some selection process, ignoring the failures that did not. The surviving sample is systematically unrepresentative, inflating estimates of success and hiding the true risk.

Question 17

When do you use a t-test versus a z-test?

Accepted Answer

Use a z-test when the population standard deviation is known and the sample is large (n >= 30, by convention); use a t-test when the standard deviation must be estimated from the sample, which is almost always the case in practice. For large n the two tests converge, but the t-test is the safe default.

Question 18

What are Type I and Type II errors, and how do they trade off?

Accepted Answer

A Type I error is rejecting a true null hypothesis (false positive), controlled by the significance level alpha. A Type II error is failing to reject a false null hypothesis (false negative), with probability beta. Lowering alpha reduces Type I errors but increases Type II errors, so the right balance depends on the cost of each mistake.

Question 19

What is the difference between variance and standard deviation, and why do we need both?

Accepted Answer

Variance is the average squared deviation from the mean; standard deviation is its square root and lives in the same units as the data. Variance is mathematically tractable — variances of independent variables add — while standard deviation is interpretable as a typical distance from the mean.

Question 20

What is a p-value, and what does it actually tell you?

Accepted Answer

A p-value is the probability of observing data at least as extreme as the collected data, assuming the null hypothesis is true. It measures surprise under H0 — not the probability that H0 is true or false.

Question 21

What is a z-score and what is standardization used for in data science?

Accepted Answer

A z-score expresses how many standard deviations an observation is from the mean of its distribution, converting raw values to a common unitless scale. Standardization — subtracting the mean and dividing by the standard deviation — is essential before algorithms that depend on distances or regularization penalties, because it prevents features with large numeric ranges from dominating those with small ranges.

Question 22

When is the mean a misleading summary statistic, and what should you use instead?

Accepted Answer

The mean is distorted by skewness and outliers, masks multimodality, and can describe a value that no individual in the dataset actually holds. Skewed, heavy-tailed, or multimodal distributions almost always require the median, percentiles, or the full distributional picture rather than the mean.

Question 23

How do you calculate sample size, statistical power, and minimum detectable effect for an A/B test?

Accepted Answer

Sample size, power, and MDE form a three-way trade-off: fix any two and the third is determined. You choose the MDE based on business materiality, then solve for the sample size that delivers 80–90 % power at alpha = 0.05.

Question 24

What is ANOVA, and why use it instead of multiple t-tests?

Accepted Answer

ANOVA (Analysis of Variance) tests whether the means of three or more groups are simultaneously equal by partitioning total variance into between-group and within-group components. Running multiple pairwise t-tests instead inflates the family-wise Type I error rate, which ANOVA avoids by using a single omnibus F-test.

Question 25

Walk me through Bayes' theorem with a disease-screening base-rate example.

Accepted Answer

Bayes' theorem updates a prior probability with new evidence: P(H|E) = P(E|H) P(H) / P(E). In disease testing, ignoring the low base rate (prior) makes a positive test look far more alarming than it really is — most positives are false positives when the disease is rare.

Question 26

What is the birthday problem and what does it reveal about probability intuition?

Accepted Answer

In a room of just 23 people, the probability that at least two share a birthday exceeds 50 %. The counterintuitive result comes from counting the large number of pairs rather than comparing each person to a fixed date — an example of how our intuition systematically underestimates collision probabilities.

Question 27

What does the Central Limit Theorem actually say, and why does it matter?

Accepted Answer

The CLT states that the sampling distribution of the sample mean converges to a normal distribution as sample size grows, regardless of the shape of the underlying population distribution. It is the theoretical foundation for confidence intervals, hypothesis tests, and many machine-learning approximations — but it applies to the distribution of the mean, not to the raw data.

Question 28

How do you choose a primary metric and guardrail metrics for an A/B test?

Accepted Answer

The primary metric should be the closest measurable proxy for the user or business outcome you are trying to move, sensitive enough to detect a real change within your traffic budget. Guardrail metrics are the constraints — things that must not degrade even if the primary metric improves.

Question 29

When does each common distribution arise — Bernoulli, Binomial, Poisson, Normal, Exponential, Uniform?

Accepted Answer

Each distribution has a natural generative story: Bernoulli is a single coin flip; Binomial sums Bernoullis; Poisson counts rare arrivals; Normal emerges from sums of many small effects; Exponential models waiting times between Poisson events; Uniform assigns equal probability across a range. Choosing correctly comes from matching that story to the data-generating process.

Question 30

What is the correct interpretation of a 95% confidence interval?

Accepted Answer

A 95% confidence interval means that if you repeated the sampling procedure many times and built an interval each time, 95% of those intervals would contain the true parameter. It does not mean there is a 95% probability that this specific interval contains the parameter.

Question 31

What is a confounding variable, and how do you control for it?

Accepted Answer

A confounding variable is associated with both the treatment and the outcome, creating a spurious apparent relationship between them. Controlling for confounders — through randomisation, stratification, regression adjustment, or matching — is essential to recover a valid causal estimate.

Question 32

What is the memoryless property of the Exponential distribution, and why does it matter?

Accepted Answer

An Exponential random variable satisfies P(X > s+t | X > s) = P(X > t): the remaining waiting time has the same distribution regardless of how long you have already waited. This is the only continuous distribution with this property, and it directly corresponds to a constant hazard rate — neither ageing nor improving over time.

Question 33

How do you handle outliers statistically, and how do you decide whether to remove them?

Accepted Answer

Handling outliers starts with understanding whether they are errors, rare genuine observations, or leverage points that reveal real signal. The appropriate response — removal, transformation, robust estimation, or explicit modelling — depends entirely on their cause, not on how extreme they look.

Question 34

How long should you run an A/B test?

Accepted Answer

Run until you have reached the pre-calculated sample size — which should include at least one full weekly cycle to average out day-of-week effects. Stopping early because results look good, or extending because they do not, both inflate error rates.

Question 35

How do you design an A/B test from scratch?

Accepted Answer

A rigorous A/B test requires a pre-registered hypothesis, a single primary metric, sample size calculated before launch, random unit-level assignment, and a fixed runtime. Skipping any of these steps opens the door to false positives and post-hoc rationalization.

Question 36

An A/B test comes back non-significant. How do you interpret and communicate that result?

Accepted Answer

A non-significant result does not mean the treatment has no effect; it means the data are insufficient to distinguish the observed difference from noise at the specified power level. The correct interpretation depends entirely on the statistical power of the test — a well-powered flat result is evidence of no meaningful effect; an underpowered flat result is inconclusive.

Question 37

Explain joint, marginal, and conditional distributions and how to move between them.

Accepted Answer

The joint distribution P(X, Y) fully specifies two random variables together. Marginals P(X) and P(Y) are obtained by summing (or integrating) the joint over the other variable. Conditionals P(X|Y=y) are the joint sliced at a fixed y value, renormalized by the marginal P(Y=y).

Question 38

What is the Law of Large Numbers and how does it differ from the Central Limit Theorem?

Accepted Answer

The Law of Large Numbers (LLN) says the sample mean converges to the true mean as sample size grows — it is a statement about where the mean lands. The Central Limit Theorem says the sampling distribution of the mean is approximately normal — it is a statement about the shape of that distribution. LLN guarantees convergence; CLT characterises the rate and shape of that convergence.

Question 39

What are long-tailed (heavy-tailed) distributions and why do they matter in practice?

Accepted Answer

A long-tailed distribution has tail probabilities that decay much more slowly than the exponential — meaning extreme events are far more common than a Gaussian model would predict. They appear in internet traffic, wealth, natural language word frequency, and insurance claims, and they invalidate many standard statistical techniques that assume thin tails.

Question 40

Explain the Monty Hall problem and why switching wins two-thirds of the time.

Accepted Answer

In the Monty Hall problem, a host who knows where the prize is always opens an empty door — this action transfers probability mass to the remaining unopened door, making switching win with probability 2/3 and staying win only 1/3. The key is that the host's action is not random.

Question 41

What is the multiple comparisons problem, and how does Bonferroni correction address it?

Accepted Answer

Running many hypothesis tests simultaneously inflates the probability of at least one false positive far above the nominal alpha. Bonferroni correction addresses this by dividing alpha by the number of tests, guaranteeing the family-wise error rate stays at or below alpha — at the cost of reduced power per test.

Question 42

What makes the Normal distribution so central in statistics, and when does it fail?

Accepted Answer

The Normal distribution is justified by the Central Limit Theorem — averages of large i.i.d. samples converge to Normal regardless of the underlying distribution. It is fully characterized by mean and variance, enabling closed-form inference. It fails for heavy-tailed data, skewed outcomes, bounded quantities, and rare extreme events.

Question 43

What are novelty and primacy effects in A/B testing, and how do you handle them?

Accepted Answer

Novelty effect is the temporary engagement spike users show with any change simply because it is new; primacy effect is the temporary dip when users resist a change to a familiar interface. Both cause the short-term treatment effect to differ materially from the long-term steady-state effect.

Question 44

What is p-hacking and how does multiple testing inflate false-positive rates?

Accepted Answer

P-hacking is the practice of making analytic choices — selecting metrics, segments, or time windows — after seeing data, guided by which choices produce p < 0.05. Multiple testing means that even without intent, testing many hypotheses at alpha = 0.05 expects one false positive per 20 tests.

Question 45

What is the difference between a p-value and an effect size, and why can a result be statistically significant but practically meaningless?

Accepted Answer

A p-value measures the probability of seeing data at least as extreme as observed under the null hypothesis — it quantifies evidence against the null, not the magnitude of an effect. Effect size quantifies how large the difference or relationship is in meaningful units. With a large enough sample, a trivially small effect can produce an arbitrarily small p-value.

Question 46

What is the difference between a paired and an unpaired (independent samples) t-test, and when should you use each?

Accepted Answer

A paired t-test is used when each observation in one group is naturally linked to one observation in the other — same subject before and after, or matched controls. An unpaired (independent-samples) t-test is used when the two groups have no subject-level correspondence. Pairing removes between-subject variance and increases power when that variance is substantial.

Question 47

What is the difference between parametric and non-parametric tests, and when should you prefer one over the other?

Accepted Answer

Parametric tests assume the data follow a specific distribution (usually normal) and make inferences about distributional parameters like the mean. Non-parametric tests make no such distributional assumption and typically operate on ranks. Non-parametric tests are more robust but less powerful when parametric assumptions genuinely hold.

Question 48

When would you use Spearman correlation instead of Pearson correlation?

Accepted Answer

Pearson correlation measures the strength of the linear relationship between two continuous variables and is sensitive to outliers and non-normality. Spearman correlation is Pearson applied to the ranks of the data, making it appropriate for monotonic (not necessarily linear) relationships, ordinal variables, and data with outliers or heavy-tailed distributions.

Question 49

What is regression to the mean, and why does it fool analysts into seeing treatment effects that do not exist?

Accepted Answer

Regression to the mean is the statistical tendency for extreme measurements to be followed by measurements closer to the population mean, purely due to random noise — not because of any intervention. Analysts who intervene after observing an extreme value and then observe improvement often incorrectly attribute the recovery to their action.

Question 50

What are the main sampling methods and how can sampling introduce bias?

Accepted Answer

The main probability sampling methods are simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Bias enters when some units have a zero or systematically different probability of selection — as in convenience sampling, survivorship bias, or non-response bias — making the sample unrepresentative of the target population regardless of size.