Probability distributions: binomial, poisson, normal - Question Bank

1. Which of the following is NOT a characteristic of the Normal distribution?
A) It is bell-shaped.
B) It is symmetrical about its mean.
C) It has a finite range.
D) The mean, median, and mode are equal.
2. If a random variable X follows a Binomial distribution B(n, p), and n is very large while p is very small, X can be approximated by which distribution?
A) Normal distribution
B) Poisson distribution
C) Uniform distribution
D) Geometric distribution
3. A Z-score represents the number of standard deviations a particular data point is away from the mean. For a Standard Normal distribution, what is the Z-score of the mean itself?
A) 0
B) 1
C) -1
D) Cannot be determined
4. The Poisson distribution is often used to model rare events. 'Rare' in this context means:
A) The probability of the event is extremely low in any given small interval.
B) The event occurs only once.
C) The interval of observation is very small.
D) The total number of events is small.
5. The Binomial distribution is appropriate when sampling:
A) With replacement, from a large population.
B) Without replacement, from a small population.
C) With or without replacement, as long as trials are independent.
D) Continuously.
6. When calculating probabilities for a continuous distribution like the Normal distribution, we use the probability density function (PDF). What does the area under the PDF curve between two points represent?
A) The probability of a single point.
B) The probability that the random variable falls within that range.
C) The total number of possible outcomes.
D) The mean of the distribution.
7. Which distribution is characterized by parameters 'μ' (mean) and 'σ' (standard deviation)?
A) Binomial
B) Poisson
C) Normal
D) Chi-squared
8. Which distribution is characterized by parameter 'λ' (average rate)?
A) Binomial
B) Poisson
C) Normal
D) Uniform
9. Which distribution is characterized by parameters 'n' (number of trials) and 'p' (probability of success)?
A) Poisson
B) Normal
C) Binomial
D) Exponential
10. In the context of probability distributions, what is a 'random variable'?
A) A fixed numerical value.
B) A variable whose value is determined by the outcome of a random phenomenon.
C) A parameter of a distribution.
D) The probability of an event.
11. What is the relationship between the variance and the standard deviation of a random variable X?
A) Variance = Standard Deviation
B) Standard Deviation = Variance^2
C) Variance = Standard Deviation^2
D) Variance = 1 / Standard Deviation
12. A company's sales figures are normally distributed with a mean of $50,000 and a standard deviation of $10,000. What is the probability that a randomly selected day's sales will be between $40,000 and $60,000?
A) Approximately 0.68
B) Approximately 0.95
C) Approximately 0.997
D) Approximately 0.50
13. If the average number of accidents at a specific intersection per week is 3 (λ=3), what is the probability of exactly 0 accidents occurring in a given week, according to the Poisson distribution?
A) 0.050
B) 0.150
C) 0.200
D) 0.300
14. In a Binomial distribution with n=10 and p=0.5, what is the probability of getting exactly 5 successes?
A) 0.246
B) 0.500
C) 0.100
D) 0.312
15. Which of the following scenarios would be best modeled by a Normal distribution?
A) The number of successes in 5 trials.
B) The number of typos on a page.
C) The distribution of IQ scores in a large population.
D) The outcome of a single lottery draw.
16. Which of the following scenarios would be best modeled by a Poisson distribution?
A) The number of heads in 10 coin flips.
B) The number of calls received by a call center in a 10-minute interval, with an average rate of 5 calls per interval.
C) The score on a standardized test.
D) The lifespan of a light bulb.
17. Which of the following scenarios would be best modeled by a Binomial distribution?
A) The number of customers arriving at a store per hour.
B) The height of students in a class.
C) The number of defective items in a batch of 100 manufactured units, where each item has a 5% chance of being defective.
D) The time it takes for a runner to complete a marathon.
18. The Central Limit Theorem states that the sampling distribution of the sample mean will approach a Normal distribution as the sample size increases, regardless of the population distribution. This is a fundamental concept for inferential statistics. What is a key requirement for the CLT to hold?
A) The population must be normally distributed.
B) The sample size must be sufficiently large (often considered n > 30).
C) The probability of success must be constant.
D) The events must be independent.
19. Which probability distribution is often used to approximate the Binomial distribution when 'n' is large and 'p' is close to 0.5?
A) Poisson distribution
B) Standard Normal distribution
C) Normal distribution
D) Exponential distribution
20. According to the Empirical Rule, approximately what percentage of data falls within three standard deviations of the mean in a Normal distribution?
A) 68%
B) 95%
C) 99.7%
D) 100%
21. According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a Normal distribution?
A) 68%
B) 95%
C) 99.7%
D) 75%
22. The Empirical Rule (or 68-95-99.7 rule) applies to Normal distributions. Approximately what percentage of data falls within one standard deviation of the mean?
A) 68%
B) 95%
C) 99.7%
D) 50%
23. What is the symbol for a value drawn from a Standard Normal distribution?
A) X
B) μ
C) Z
D) λ
24. The Standard Normal distribution is a special case of the Normal distribution. What are its parameters?
A) Mean = 1, Standard Deviation = 1
B) Mean = 0, Standard Deviation = 1
C) Mean = 0, Standard Deviation = 0
D) Mean = 1, Standard Deviation = 0
25. What is the total area under the curve of any continuous probability distribution, including the Normal distribution?
A) 0
B) 1
C) Infinity
D) Depends on the parameters
26. The Normal distribution is symmetrical. What does this symmetry imply about the mean, median, and mode?
A) They are all different.
B) They are all equal.
C) The mean equals the mode, but the median is different.
D) The median equals the mode, but the mean is different.
27. In a Normal distribution, what does the standard deviation (σ) measure?
A) The center of the distribution.
B) The spread or dispersion of the data around the mean.
C) The total area under the curve.
D) The number of trials.
28. What is the symbol for the standard deviation of a Normal distribution?
A) σ
B) μ
C) λ
D) p
29. What is the symbol for the mean of a Normal distribution?
A) σ
B) μ
C) λ
D) p
30. The Normal distribution is a continuous probability distribution characterized by its bell-shaped curve. What are the two main parameters that define a Normal distribution?
A) n and p
B) λ and x
C) Mean (μ) and standard deviation (σ)
D) Minimum and maximum values
31. When can the Poisson distribution be used as an approximation to the Binomial distribution?
A) When n is small and p is large.
B) When n is large and p is small (such that np is a moderate value).
C) When n is small and p is small.
D) When n is large and p is large.
32. For a Poisson distribution, what is the variance?
A) λ
B) λ^2
C) e^λ
D) 1/λ
33. For a Poisson distribution, what is the mean (expected value)?
A) λ
B) λ^2
C) e^λ
D) 1/λ
34. What is a key assumption for using the Poisson distribution?
A) The probability of an event changes significantly with the interval length.
B) Events occur independently of each other.
C) There are only two possible outcomes for each event.
D) The number of events is fixed.
35. The formula for the probability mass function (PMF) of a Poisson distribution is P(X=x) = (e^-λ * λ^x) / x!. What does 'e' represent in this formula?
A) The number of events.
B) The average rate.
C) Euler's number (approximately 2.71828).
D) The factorial symbol.
36. In the Poisson distribution, what does 'x' or 'k' typically represent?
A) The average rate of occurrence.
B) The number of events occurring in the interval.
C) The length of the interval.
D) The probability of an event.
37. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence. What is the average rate of occurrence denoted by?
A) k
B) x
C) λ (lambda)
D) e
38. What is the variance of a Binomial distribution?
A) np
B) npq
C) n
D) p
39. What is the mean (expected value) of a Binomial distribution?
A) npq
B) n/p
C) np
D) sqrt(npq)
40. Which condition must be met for a Binomial distribution to be a good approximation for a real-world scenario?
A) The number of trials must be infinite.
B) Each trial must have more than two possible outcomes.
C) The trials must be independent, and the probability of success must be constant for each trial.
D) The outcomes must be continuous.
41. The formula for the probability mass function (PMF) of a Binomial distribution is P(X=k) = C(n, k) * p^k * q^(n-k). What does C(n, k) represent?
A) The probability of failure.
B) The number of trials.
C) The number of combinations of n items taken k at a time.
D) The exponent of the success probability.
42. What does 'k' represent in the context of a Binomial distribution?
A) The number of trials.
B) The number of successes.
C) The probability of success.
D) The probability of failure.
43. What does 'n' represent in the context of a Binomial distribution?
A) The number of successes.
B) The number of trials.
C) The probability of success.
D) The probability of failure.
44. In a Binomial distribution, if 'p' is the probability of success, what is the probability of failure denoted by?
A) n
B) k
C) p
D) q
45. The Binomial distribution is used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. What is the probability of success in a single trial denoted by?
A) n
B) k
C) p
D) q
46. Which of the following is a key characteristic of a continuous probability distribution?
A) It involves countable outcomes.
B) The probability of any single exact value is zero.
C) It is defined for a finite number of outcomes.
D) The probability mass function is used to determine probabilities.
47. Which of the following is a key characteristic of a discrete probability distribution?
A) It can take any value within a given range.
B) The probabilities sum up to exactly 1 over an infinite number of possible outcomes.
C) It deals with outcomes that can only take specific, separate values.
D) It is represented by a continuous curve.
48. In probability theory, what does a 'distribution' represent?
A) A single possible outcome of an experiment.
B) The likelihood of a specific event occurring.
C) A function that describes the possible values a random variable can take and their associated probabilities.
D) The average value of a set of data.
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