Probability distributions: binomial, poisson, normal - One Line Questions

1. What is the total area under the curve of any continuous probability distribution, including the Normal distribution? — 1
2. 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? — 0
3. 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? — 0.050
4. In a Binomial distribution with n=10 and p=0.5, what is the probability of getting exactly 5 successes? — 0.246
5. 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? — 68%
6. According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a Normal distribution? — 95%
7. According to the Empirical Rule, approximately what percentage of data falls within three standard deviations of the mean in a Normal distribution? — 99.7%
8. In the context of probability distributions, what is a 'random variable'? — A variable whose value is determined by the outcome of a random phenomenon.
9. In probability theory, what does a 'distribution' represent? — A function that describes the possible values a random variable can take and their associated probabilities.
10. 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? — Approximately 0.68
11. Which distribution is characterized by parameter 'λ' (average rate)? — Poisson
12. Which distribution is characterized by parameters 'μ' (mean) and 'σ' (standard deviation)? — Normal
13. Which of the following is a key characteristic of a discrete probability distribution? — It deals with outcomes that can only take specific, separate values.
14. Which of the following is a key characteristic of a continuous probability distribution? — The probability of any single exact value is zero.
15. Which of the following is NOT a characteristic of the Normal distribution? — It has a finite range.
16. 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? — λ (lambda)
17. The Standard Normal distribution is a special case of the Normal distribution. What are its parameters? — Mean = 0, Standard Deviation = 1
18. 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? — p
19. In a Binomial distribution, if 'p' is the probability of success, what is the probability of failure denoted by? — q
20. 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? — Mean (μ) and standard deviation (σ)
21. 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? — Poisson distribution
22. What is the variance of a Binomial distribution? — npq
23. What is the mean (expected value) of a Binomial distribution? — np
24. Which distribution is characterized by parameters 'n' (number of trials) and 'p' (probability of success)? — Binomial
25. Which probability distribution is often used to approximate the Binomial distribution when 'n' is large and 'p' is close to 0.5? — Normal distribution
26. In the Poisson distribution, what does 'x' or 'k' typically represent? — The number of events occurring in the interval.
27. In a Normal distribution, what does the standard deviation (σ) measure? — The spread or dispersion of the data around the mean.
28. Which of the following scenarios would be best modeled by a Binomial distribution? — The number of defective items in a batch of 100 manufactured units, where each item has a 5% chance of being defective.
29. 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? — Euler's number (approximately 2.71828).
30. Which of the following scenarios would be best modeled by a Poisson distribution? — The number of calls received by a call center in a 10-minute interval, with an average rate of 5 calls per interval.
31. Which of the following scenarios would be best modeled by a Normal distribution? — The distribution of IQ scores in a large population.
32. What does 'n' represent in the context of a Binomial distribution? — The number of trials.
33. Which condition must be met for a Binomial distribution to be a good approximation for a real-world scenario? — The trials must be independent, and the probability of success must be constant for each trial.
34. What does 'k' represent in the context of a Binomial distribution? — The number of successes.
35. 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? — The sample size must be sufficiently large (often considered n > 30).
36. 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? — The probability that the random variable falls within that range.
37. What is a key assumption for using the Poisson distribution? — Events occur independently of each other.
38. 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? — The number of combinations of n items taken k at a time.
39. The Poisson distribution is often used to model rare events. 'Rare' in this context means: — The probability of the event is extremely low in any given small interval.
40. The Normal distribution is symmetrical. What does this symmetry imply about the mean, median, and mode? — They are all equal.
41. What is the relationship between the variance and the standard deviation of a random variable X? — Variance = Standard Deviation^2
42. When can the Poisson distribution be used as an approximation to the Binomial distribution? — When n is large and p is small (such that np is a moderate value).
43. The Binomial distribution is appropriate when sampling: — With replacement, from a large population.
44. What is the symbol for a value drawn from a Standard Normal distribution? — Z
45. For a Poisson distribution, what is the mean (expected value)? — λ
46. For a Poisson distribution, what is the variance? — λ
47. What is the symbol for the mean of a Normal distribution? — μ
48. What is the symbol for the standard deviation of a Normal distribution? — σ
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