This is the kind of question that looks easy on paper and then you second-guess yourself mid-derivation.
Start by writing the likelihood function for an i.i.d. Normal sample, then take the log and derive the partial derivatives with respect to μ and σ². Set the derivatives to zero and solve to obtain the MLEs, noting that the variance estimator is biased and uses n in the denominator.
Pro tip: Mention that the MLE for variance is biased and that the unbiased estimator uses n-1, showing awareness of practical implications. Also, connect the MLEs to the method of moments and discuss their asymptotic properties like consistency and efficiency.
For an i.i.d. sample x₁, ..., xₙ from N(μ, σ²), the joint density is the product of individual normal densities. Write L(μ, σ²) = ∏ (1/√(2πσ²)) exp(-(xᵢ - μ)²/(2σ²)).
Simplify by taking the natural logarithm: ℓ(μ, σ²) = -n/2 log(2π) - n/2 log(σ²) - (1/(2σ²)) Σ (xᵢ - μ)².
Compute ∂ℓ/∂μ and ∂ℓ/∂σ². For μ: ∂ℓ/∂μ = (1/σ²) Σ (xᵢ - μ). For σ²: ∂ℓ/∂σ² = -n/(2σ²) + (1/(2σ⁴)) Σ (xᵢ - μ)².
Set ∂ℓ/∂μ = 0 to get μ̂ = (1/n) Σ xᵢ = x̄. Set ∂ℓ/∂σ² = 0 and substitute μ̂ to get σ̂² = (1/n) Σ (xᵢ - x̄)².
Check second derivatives for maximum. Note that μ̂ is unbiased, but σ̂² is biased; the unbiased estimator is s² = (1/(n-1)) Σ (xᵢ - x̄)². Mention consistency and asymptotic normality.
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Straightforward if you remember the formula.
Start by clearly defining the conditional distribution formula for continuous random variables: f_{X|Y}(x|y) = f_{X,Y}(x,y) / f_Y(y), where f_Y(y) is the marginal pdf of Y. Then explain how to obtain the marginal pdf by integrating the joint pdf over all possible values of X, and finally discuss the importance of the support and any assumptions (e.g., f_Y(y) > 0).
Pro tip: Emphasize that the conditional distribution is only defined when the marginal pdf of Y is positive at the given value, and mention that in practice, you might need to handle cases where the joint pdf is zero or the marginal is zero. This shows attention to mathematical rigor and practical edge cases.
Write the formal definition of the conditional pdf: f_{X|Y}(x|y) = f_{X,Y}(x,y) / f_Y(y), for f_Y(y) > 0.
Integrate the joint pdf over the entire range of X: f_Y(y) = ∫ f_{X,Y}(x,y) dx. Ensure the integration limits are correct based on the support of X.
Plug the marginal pdf into the conditional formula and simplify the expression if possible. Specify the support of the conditional distribution.
Check that the resulting conditional pdf integrates to 1 over x for any fixed y, and discuss any conditions or edge cases (e.g., when f_Y(y)=0).
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I blanked on the exact constant in the pdf for a second.
Start by clearly stating the PDF and CDF formulas for the standard normal distribution, using proper notation. Then briefly explain the parameters (mean 0, variance 1) and mention that the CDF is the integral of the PDF. Finally, connect these to practical applications in data science, such as hypothesis testing and confidence intervals.
Pro tip: Emphasize that the standard normal is a special case of the normal distribution with μ=0 and σ=1, and mention that the CDF is often denoted by Φ(z). This shows you understand the notation and can communicate precisely.
State that it's a normal distribution with mean 0 and variance 1, denoted as Z ~ N(0,1).
Present the probability density function: f(z) = (1/√(2π)) * e^(-z²/2). Explain each component briefly.
Present the cumulative distribution function: Φ(z) = ∫_{-∞}^{z} (1/√(2π)) * e^(-t²/2) dt. Note that it has no closed-form expression in elementary functions.
Mention symmetry, the 68-95-99.7 rule, and how the CDF is used to compute p-values and critical values in statistical tests.
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This one surprised me a bit because it's more conceptual than mechanical.
Start with a concrete estimation problem, such as estimating the average treatment effect in an A/B test. Then, systematically outline the assumptions required for an estimator to be unbiased, explaining each assumption's role and how to justify it in practice. Emphasize the importance of validating assumptions through data diagnostics and domain knowledge.
Pro tip: Connect each assumption to potential real-world violations and their impact on bias, showing you can anticipate and mitigate issues like selection bias or measurement error. This demonstrates practical maturity beyond textbook definitions.
Select a relevant problem, e.g., estimating the mean of a population or the average treatment effect in an A/B test. Clearly define the estimand and the estimator.
Identify assumptions such as random sampling, independence, correct model specification, no measurement error, and exogeneity. Explain why each is needed for the estimator to be unbiased.
For each assumption, discuss how it can be justified using domain knowledge, study design (e.g., randomization), or statistical tests. Mention consequences if violated.
Explain how violations lead to bias and what trade-offs exist (e.g., bias vs. variance). Highlight methods to detect and correct violations, such as robust estimators or sensitivity analysis.
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