Probability theory: concepts, distributions, moments and central limit theorem. - One Line Questions

1. What is the probability of a certain event? — 1
2. What is the probability of an impossible event? — 0
3. What is the minimum sample size generally considered 'sufficiently large' for the Central Limit Theorem to apply reasonably well? — 30
4. A random variable is a function that assigns: — A numerical value to each outcome of a random experiment.
5. In probability theory, an 'event' refers to: — A collection of outcomes of a random experiment.
6. What is the sample space of a random experiment? — The set of all possible outcomes of the experiment.
7. A discrete random variable can take values that are: — A finite or countably infinite number of distinct values.
8. Which theorem is fundamental for statistical inference, allowing us to approximate complex distributions with the normal distribution for large samples? — Central Limit Theorem
9. The CLT is crucial in statistics because it allows us to: — Make inferences about population parameters using sample statistics, even if the population distribution is unknown.
10. The probability mass function (PMF) is associated with which type of random variable? — Discrete
11. The probability density function (PDF) is associated with which type of random variable? — Continuous
12. The variance is defined as: — E[(X - E(X))^2]
13. According to the Central Limit Theorem, what is the mean of the sampling distribution of the sample mean? — Equal to the population mean (μ).
14. For a continuous random variable X, P(X = c) is always: — Equal to 0
15. What is the primary characteristic of a continuous uniform distribution? — All outcomes within a given range are equally likely.
16. Which of the following is a property of the CDF, F(x)? — It is non-decreasing and ranges from 0 to 1.
17. If a distribution has a skewness of 0, it implies: — It is symmetric.
18. Which of the following is NOT a moment of a probability distribution? — Mean (First moment about the origin)
19. A standard Normal distribution has: — Mean = 0 and Standard Deviation = 1
20. The Normal distribution is a continuous probability distribution that is symmetric around its mean and is characterized by two parameters: — Mean (μ) and Variance (σ^2)
21. The Poisson distribution is often used to approximate the Binomial distribution when: — n is large and p is small.
22. Which distribution is commonly used to model the number of successes in a fixed number of trials, where each trial has only two outcomes? — Binomial Distribution
23. A continuous random variable can take values that are: — Any value within a given interval or range.
24. The concept of 'moments' in probability theory helps describe the shape of a distribution by quantifying: — The central tendency, spread, skewness, and kurtosis.
25. The conditional probability P(A|B) is defined as: — P(A and B) / P(B)
26. If events A and B are independent, then: — P(A and B) = P(A) * P(B)
27. If P(A) denotes the probability of event A, which of the following is always true? — 0 <= P(A) <= 1
28. Bayes' Theorem relates conditional probabilities and is stated as: — P(A|B) = P(B|A) * P(A) / P(B)
29. Skewness measures the: — Asymmetry or lack of symmetry of the distribution.
30. Which distribution is characterized by two parameters: n (number of trials) and p (probability of success)? — Binomial distribution
31. According to the Central Limit Theorem, what is the standard deviation of the sampling distribution of the sample mean (also known as the standard error)? — Population standard deviation divided by the square root of the sample size (σ/√n).
32. The Central Limit Theorem (CLT) states that the distribution of the sample mean approaches a normal distribution: — Regardless of the population distribution, if the sample size is sufficiently large.
33. The shape of the Normal distribution is: — Bell-shaped and symmetric
34. For a discrete random variable X with PMF f(x), the expected value is calculated as: — Sum of x * f(x) over all possible x.
35. Kurtosis measures the: — Peakedness and tail heaviness of the distribution.
36. The variance of a random variable X, Var(X) or sigma^2, measures: — The spread or dispersion of the values around the mean.
37. The second moment about the mean is: — The variance.
38. The Binomial distribution is suitable for modeling: — The number of successes in a fixed number of independent Bernoulli trials.
39. The exponential distribution is often used to model: — The time until the next event occurs in a Poisson process.
40. The Poisson distribution is characterized by a single parameter, lambda (λ), which represents: — The average rate of occurrence of an event.
41. Two events A and B are considered mutually exclusive if: — They cannot occur at the same time; P(A and B) = 0.
42. What is the 'Law of Large Numbers' related to probability theory? — The average of the results obtained from a large number of trials should be close to the expected value.
43. The cumulative distribution function (CDF), F(x), represents: — The probability of X being less than or equal to x, i.e., P(X <= x).
44. What is the fundamental concept of probability theory? — The quantification and study of randomness and uncertainty.
45. The standard deviation is: — The square root of the variance.
46. The first moment about the origin of a random variable X is: — The expected value E(X).
47. A distribution with high kurtosis (leptokurtic) tends to have: — Heavy tails and a sharp peak.
48. The expected value of a random variable X, denoted E(X), is also known as the: — Mean or Average Value
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