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Distribution Properties Practice

Test your understanding of common distribution families, exponential family theory, and statistical inference with 15 comprehensive questions

15
Total Questions
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Questions Attempted
Statistical Inference
Questions related to statistical inference concepts and applications
Question 1: Point Estimation
In statistical inference, using a single numerical value to estimate a population parameter is calle...
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Sample Statistics
Questions related to sample statistics concepts and applications
Question 2: Sample Mean Variance
When the population has finite second moment with E(X) = μ and Var(X) = σ², the variance of the samp...
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Normal Distribution
Questions related to normal distribution concepts and applications
Question 3: Normal Sample Mean Distribution
If X₁, X₂, ..., Xₙ is a simple random sample from N(μ, σ²), then the sample mean X̄ follows:...
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Question 13: Normal Standardization
If X ~ N(μ, σ²), then (X - μ)/σ follows:...
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Gamma Distribution
Questions related to gamma distribution concepts and applications
Question 4: Gamma Distribution Additivity
If X₁ ~ Γ(α₁, λ) and X₂ ~ Γ(α₂, λ) are independent, then X₁ + X₂ follows:...
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Question 11: Gamma Special Cases
The relationship between Gamma and Exponential distributions is:...
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Chi-Square Distribution
Questions related to chi-square distribution concepts and applications
Question 5: Chi-Square Construction
If X₁, X₂, ..., Xₙ are independent and identically distributed as N(0,1), then ∑ᵢ₌₁ⁿ Xᵢ² follows:...
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Question 12: Chi-Square Additivity
If ξ₁ ~ χ²(n₁) and ξ₂ ~ χ²(n₂) are independent, then ξ₁ + ξ₂ follows:...
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t-Distribution
Questions related to t-distribution concepts and applications
Question 6: t-Distribution Limit
As n → ∞, the t(n) distribution:...
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Question 10: t-Distribution Construction
If X ~ N(0,1) and K ~ χ²(n) are independent, then T = X/√(K/n) follows:...
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F-Distribution
Questions related to f-distribution concepts and applications
Question 7: F-Distribution Definition
If K₁ ~ χ²(m) and K₂ ~ χ²(n) are independent, then F = (K₁/m)/(K₂/n) follows:...
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Question 9: F-Distribution Reciprocal Property
For F-distribution, if F ~ F(m,n), then:...
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Exponential Family
Questions related to exponential family concepts and applications
Question 8: Non-Exponential Family
Among the following distribution families, which one does NOT belong to the exponential family:...
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Question 14: Exponential Family Form
The general form of exponential family density is:...
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Distribution Properties
Questions related to distribution properties concepts and applications
Question 15: Sufficient Statistics
In exponential families, sufficient statistics have the property that:...
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