Binomial, normal and Poisson distributions – properties and examples MCQs With Answer

Introduction: Understanding probability distributions is essential for B. Pharm students who analyze pharmaceutical data. This concise guide covers the binomial, normal and Poisson distributions — their definitions, parameters (mean, variance), probability mass/density functions, properties, assumptions and practical pharmaceutical examples such as defect counts, stability results, assay variability and microbial colony counts. You will learn when to use binomial for fixed-trial success/failure data, Poisson for rare counts and normal for continuous measurements and approximations via the central limit theorem. Emphasis is on parameter estimation, approximation rules and applications in quality control and regulatory reporting. Now let’s test your knowledge with 30 MCQs on this topic.

Q1. Which statement best defines a binomial distribution?

• A distribution for counts of events occurring in a fixed time interval with mean λ
• A distribution of continuous measurements that is symmetric around the mean
• A distribution for the number of successes in a fixed number of independent trials with constant success probability
• A distribution for time-to-event data with a constant hazard

Correct Answer: A distribution for the number of successes in a fixed number of independent trials with constant success probability

Q2. What are the mean and variance of a binomial(n, p) distribution?

• Mean = np, Variance = np(1 − p)
• Mean = p, Variance = p(1 − p)/n
• Mean = n/p, Variance = np
• Mean = λ, Variance = λ

Correct Answer: Mean = np, Variance = np(1 − p)

Q3. For a Poisson(λ) distribution which property holds true?

• Mean = np and Variance = np(1 − p)
• Mean = λ and Variance = λ
• Mean = 0 and Variance = 1
• Mean = μ and Variance = σ² independent of λ

Correct Answer: Mean = λ and Variance = λ

Q4. Which formula represents the Poisson probability P(X = k)?

• P(X = k) = C(n,k) p^k (1 − p)^(n−k)
• P(X = k) = (1/√(2πσ²)) e^{−(x−μ)²/(2σ²)}
• P(X = k) = e^{−λ} λ^k / k!
• P(X = k) = λ e^{−λ x}

Correct Answer: P(X = k) = e^{−λ} λ^k / k!

Q5. Which description fits the normal distribution?

• A discrete distribution for count data often used for rare events
• A continuous, symmetric, bell-shaped distribution fully described by mean and variance
• A skewed distribution defined only for nonnegative values
• A distribution with mean equal to variance equal to λ

Correct Answer: A continuous, symmetric, bell-shaped distribution fully described by mean and variance

Q6. How do you transform a normal random variable X ~ N(μ, σ²) to the standard normal Z?

• Z = (X + μ)/σ
• Z = (X − μ)/σ
• Z = Xσ + μ
• Z = (X − μ)/σ²

Correct Answer: Z = (X − μ)/σ

Q7. Which of the following is NOT a requirement for a binomial model?

• A fixed number of trials
• Each trial has two outcomes: success or failure
• Probability of success is allowed to change between trials
• Trials are independent

Correct Answer: Probability of success is allowed to change between trials

Q8. When is a Poisson model appropriate in pharmaceutical contexts?

• For measuring continuous assay values that are symmetric
• For counting rare events (e.g., impurities per unit) when events are independent and occur at a low average rate
• For the proportion of tablets passing dissolution in a fixed sample
• For modeling time-to-failure assuming a normal error structure

Correct Answer: For counting rare events (e.g., impurities per unit) when events are independent and occur at a low average rate

Q9. Which rule-of-thumb is commonly used for the normal approximation to a binomial?

• Use when n is small and p is very close to 0 or 1
• Use when both np ≥ 5 and n(1 − p) ≥ 5
• Use only when p = 0.5
• Never use normal approximation for binomial

Correct Answer: Use when both np ≥ 5 and n(1 − p) ≥ 5

Q10. What is the continuity correction when approximating a discrete binomial probability by a normal distribution?

• Add or subtract 0.5 to the discrete x value before standardizing
• Multiply the mean by 0.5
• Use variance = np instead of np(1 − p)
• Round probabilities to the nearest 0.1

Correct Answer: Add or subtract 0.5 to the discrete x value before standardizing

Q11. The central limit theorem implies which result about sample means?

• Sample means are exactly normal for any sample size
• For large n, the sampling distribution of the sample mean is approximately normal regardless of the parent distribution
• Sampling distribution variance increases with n
• Only binomial parent populations yield normal sample means

Correct Answer: For large n, the sampling distribution of the sample mean is approximately normal regardless of the parent distribution

Q12. How is the Poisson parameter λ commonly estimated from data?

• λ is estimated as the sample variance multiplied by n
• λ is estimated as the sample mean of counts
• λ is estimated as p/(1 − p)
• λ is fixed and cannot be estimated

Correct Answer: λ is estimated as the sample mean of counts

Q13. For a binomial distribution, how does variance compare to the mean?

• Variance is always greater than the mean
• Variance equals the mean
• Variance is less than or equal to the mean (variance = mean × (1 − p))
• Variance is independent and unrelated to the mean

Correct Answer: Variance is less than or equal to the mean (variance = mean × (1 − p))

Q14. Which distribution would you typically assume for repeated assay measurements of a drug concentration?

• Binomial distribution
• Poisson distribution
• Normal distribution
• Exponential distribution

Correct Answer: Normal distribution

Q15. Which scenario is a classic application of the Poisson distribution in pharmaceutics?

• Proportion of tablets failing dissolution in a 20-sample test
• Number of bacterial colonies on agar per plate
• Average potency (continuous) across batches
• Time until tablet disintegration with normal errors

Correct Answer: Number of bacterial colonies on agar per plate

Q16. Under what limit does a binomial(n, p) distribution converge to a Poisson distribution?

• As n → ∞ and p → 0 with np = λ fixed
• As p → 1 and n fixed
• When n is small and p is moderate
• Never — they are unrelated

Correct Answer: As n → ∞ and p → 0 with np = λ fixed

Q17. Which of the following distributions are discrete?

• Normal and Poisson
• Binomial and Normal
• Binomial and Poisson
• All three: Binomial, Normal, Poisson

Correct Answer: Binomial and Poisson

Q18. If events occur at a constant average rate λ per hour, what is the Poisson probability of zero events in t hours?

• 1 − e^{−λt}
• e^{−λt}
• λt e^{−λt}
• 0

Correct Answer: e^{−λt}

Q19. Approximately what proportion of values lie within ±1 standard deviation of the mean in a normal distribution?

• About 50%
• About 68%
• About 95%
• About 99.7%

Correct Answer: About 68%

Q20. A sample observation has Z = 2 in a standard normal. Approximately what percentile is this observation?

• About 50th percentile
• About 84th percentile
• About 97.5th percentile
• About 99.9th percentile

Correct Answer: About 97.5th percentile

Q21. How does the standard error of the sample mean change with sample size n?

• It increases proportionally to √n
• It decreases proportionally to 1/√n
• It stays constant regardless of n
• It equals σ²/n

Correct Answer: It decreases proportionally to 1/√n

Q22. If the probability of a defective tablet is 0.02 in a batch of n = 100, what is the expected number of defectives?

• 0.02
• 2
• 50
• 98

Correct Answer: 2

Q23. Which test commonly uses the normal approximation for proportions in large samples?

• Chi-squared test only
• Z-test for proportions
• F-test for variances
• Wilcoxon signed-rank test

Correct Answer: Z-test for proportions

Q24. How would you detect overdispersion in count data when considering a Poisson model?

• Check if variance ≈ mean; overdispersion if variance < mean
• Check if variance ≈ mean; overdispersion if variance > mean
• Overdispersion is irrelevant for Poisson
• Overdispersion exists if mean = 0

Correct Answer: Check if variance ≈ mean; overdispersion if variance > mean

Q25. What is the binomial probability mass function?

• P(X = k) = e^{−λ} λ^k / k!
• P(X = k) = (1/√(2πσ²)) e^{−(x−μ)²/(2σ²)}
• P(X = k) = C(n,k) p^k (1 − p)^{n−k}
• P(X = k) = λ e^{−λ x}

Correct Answer: P(X = k) = C(n,k) p^k (1 − p)^{n−k}

Q26. For X ~ N(50, 5²), what is approximately P(X > 60)?

• About 0.50
• About 0.1587
• About 0.0228
• About 0.0013

Correct Answer: About 0.0228

Q27. The standard error of a sample proportion p̂ is given by which formula?

• √[p̂(1 − p̂)/n]
• p̂(1 − p̂)
• √(np̂(1 − p̂))
• p̂/√n

Correct Answer: √[p̂(1 − p̂)/n]

Q28. Counting the number of impurities per vial over many vials is best modeled by which distribution?

• Normal distribution
• Binomial distribution with small n
• Poisson distribution
• Uniform distribution

Correct Answer: Poisson distribution

Q29. For a binomial with n = 100 and p = 0.01, which approximation is most appropriate?

• Normal approximation without correction
• Poisson approximation with λ = 1
• Exponential approximation
• Use uniform distribution approximation

Correct Answer: Poisson approximation with λ = 1

Q30. Which statement about approximations is true?

• Poisson can approximate binomial when p is large
• Normal can approximate Poisson when λ is small (λ < 1)
• Normal approximation to Poisson is reasonable when λ is large (e.g., λ ≥ 10)
• Binomial approximates normal only if p = 0.5

Correct Answer: Normal approximation to Poisson is reasonable when λ is large (e.g., λ ≥ 10)

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