Question:
A research team studies the probability of crop damage by wild boar in crop fields. For each crop field sampled, they record 1 if damage was observed, and 0 if damage was not observed. Which one of the following distributions is most appropriate to analyse the probability of crop damage?
A research team studies the probability of crop damage by wild boar in crop fields. For each crop field sampled, they record 1 if damage was observed, and 0 if damage was not observed. Which one of the following distributions is most appropriate to analyse the probability of crop damage?
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Binary outcome per trial \Rightarrow use \textbf{Bernoulli} for one trial and \textbf{Binomial} for the sum over \(n\) independent, identical trials. Poisson is for counts with no fixed number of trials.
Updated On: Aug 27, 2025
- Binomial distribution
- Poisson distribution
- Cauchy distribution
- Gamma distribution
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The Correct Option is A
Solution and Explanation
Step 1: Recognize the data type.
Each field produces a binary outcome: damaged \(=1\) or not damaged \(=0\). For a single field this is a Bernoulli trial with success probability \(p=\Pr(\text{damage})\).
Step 2: Aggregate across many fields.
If \(n\) independent fields are observed under similar conditions, the total number of damaged fields \(X\) follows \[ X \sim \text{Binomial}(n,p), \] because \(X\) is the sum of \(n\) independent Bernoulli(\(p\)) variables. Inference for \(p\) (including proportion estimates and logistic models) uses the binomial likelihood.
Step 3: Eliminate other options.
Poisson models counts of events in continuous space/time with rate parameter; not appropriate for fixed-\(n\) binary trials.
Cauchy is a continuous distribution with heavy tails; irrelevant here.
Gamma is continuous on \((0,\infty)\); used for waiting times or positive continuous quantities, not binary outcomes. Final Answer:
\[ \boxed{\text{(A) Binomial distribution}} \]
Each field produces a binary outcome: damaged \(=1\) or not damaged \(=0\). For a single field this is a Bernoulli trial with success probability \(p=\Pr(\text{damage})\).
Step 2: Aggregate across many fields.
If \(n\) independent fields are observed under similar conditions, the total number of damaged fields \(X\) follows \[ X \sim \text{Binomial}(n,p), \] because \(X\) is the sum of \(n\) independent Bernoulli(\(p\)) variables. Inference for \(p\) (including proportion estimates and logistic models) uses the binomial likelihood.
Step 3: Eliminate other options.
Poisson models counts of events in continuous space/time with rate parameter; not appropriate for fixed-\(n\) binary trials.
Cauchy is a continuous distribution with heavy tails; irrelevant here.
Gamma is continuous on \((0,\infty)\); used for waiting times or positive continuous quantities, not binary outcomes. Final Answer:
\[ \boxed{\text{(A) Binomial distribution}} \]
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