Probability distributions - binomial, Poisson, normal, gamma, beta, Cauchy, multinomial, hypergeometric, negative binomial - One Line Questions

1. The median of a standard Normal distribution (mu=0, sigma=1) is: — 0
2. In a binomial experiment, if n=10 and p=0.5, what is the probability of getting exactly 5 successes? — 0.246
3. In the context of the Beta distribution, if alpha = 1 and beta = 1, what distribution does it represent? — A uniform distribution on [0, 1]
4. The Gamma distribution is a generalization of the exponential distribution. If the shape parameter k=1, the Gamma distribution becomes: — An Exponential distribution
5. The Multinomial distribution is a generalization of the binomial distribution. It describes the outcome of: — A fixed number of independent trials, each with more than two possible outcomes
6. A single draw from a Multinomial distribution with parameters n and p1, p2, ..., pk results in: — A vector of k counts, summing to n, representing the number of times each outcome occurred.
7. Consider a sequence of Bernoulli trials. The Negative Binomial distribution models the probability of obtaining: — A specific number of failures before the k-th success.
8. Which distribution is characterized by the probability of k successes in n trials, where sampling is done WITHOUT replacement from a finite population of size N containing K successes? — Hypergeometric
9. When sampling from a finite population without replacement, if we are interested in the number of successes in a fixed number of draws, which distribution is appropriate? — Hypergeometric distribution
10. If a die is rolled 10 times, and we are interested in the number of times each face (1 through 6) appears, which distribution would model this scenario? — Multinomial distribution
11. A box contains 10 red balls and 5 blue balls. If 3 balls are drawn without replacement, the probability distribution for the number of red balls drawn follows: — Hypergeometric distribution
12. The Beta distribution is related to the Gamma distribution through the Beta function B(x, y) = Gamma(x)Gamma(y) / Gamma(x+y). This relationship is useful for: — Both A and B.
13. The Cauchy distribution's heavy tails mean that: — Extreme values are more likely than in a normal distribution.
14. Which continuous probability distribution is bell-shaped, symmetric, and characterized by its mean (mu) and standard deviation (sigma)? — Normal distribution
15. Which distribution is characterized by the probability density function f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-0.5 * ((x - mu) / sigma)^2)? — Normal distribution
16. The Cauchy distribution has a characteristic function (phi(t)) given by exp(|t|). This implies: — It has heavy tails.
17. Which of the following is a property of the Normal distribution? — It is symmetric about its mean.
18. The variance of a Poisson distribution with parameter lambda is: — lambda
19. The Cauchy distribution is known for its heavy tails and lack of a well-defined mean or variance. What is its characteristic parameter? — Mu (location) and Sigma (scale)
20. The probability density function of a Poisson distribution with parameter lambda is given by P(X=k) = (lambda^k * e^(-lambda)) / k! for k = 0, 1, 2, ... . This formula is valid when: — lambda > 0.
21. If X follows a Poisson distribution with parameter lambda, what is the expected value (mean) of X? — lambda
22. What are the parameters of a Beta distribution, typically denoted by alpha and beta? — Two positive shape parameters
23. In a standard normal distribution, what are the values of the mean and standard deviation? — Mean = 0, Standard Deviation = 1
24. What are the parameters of a Gamma distribution, typically denoted by k (shape) and theta (scale)? — Shape and Scale
25. The probability density function of the Cauchy distribution is f(x; x0, gamma) = 1 / (pi * gamma * [1 + ((x - x0)/gamma)^2]). Here, x0 and gamma represent: — Location and Scale
26. The Poisson distribution is often used to model the number of events occurring in a fixed interval of time or space, provided these events occur with a known average rate and independently of the time since the last event. What parameter characterizes the Poisson distribution? — lambda (average rate of occurrence)
27. Which parameters define a Hypergeometric distribution? — N (population size), K (number of success states in population), and n (number of draws)
28. The Gamma function, denoted by Gamma(z), is fundamental to the Gamma distribution. What is Gamma(n) for a positive integer n? — (n-1)!
29. When the number of trials 'n' in a binomial distribution becomes very large and the probability of success 'p' becomes very small, such that np = lambda (a constant), the binomial distribution can be approximated by which other distribution? — Poisson distribution
30. Which distribution is characterized by the fact that the sum of two independent Gamma random variables with parameters (k1, theta) and (k2, theta) is a Gamma random variable with parameter (k1+k2, theta)? — Gamma distribution
31. The sum of 'n' independent and identically distributed standard normal random variables follows a: — Chi-squared distribution with n degrees of freedom
32. For a binomial distribution B(n, p), what is the variance? — np(1-p)
33. The Normal distribution can be used to approximate the Binomial distribution when 'n' is large and 'p' is not too close to 0 or 1. What condition is typically used to determine if the approximation is valid? — np > 5 and n(1-p) > 5
34. Which probability distribution is characterized by a fixed number of independent trials, each with two possible outcomes (success or failure), and a constant probability of success? — Binomial distribution
35. The limiting distribution of the sum of a large number of independent and identically distributed random variables (under certain conditions) is the: — Normal distribution
36. What is a common parameterization for the Negative Binomial distribution, often denoted by r and p? — r = number of successes, p = probability of success
37. Which of the following statements about the Normal distribution is FALSE? — The total area under the curve is infinite.
38. The Cauchy distribution is sometimes called the 'Lorentz distribution'. A key characteristic is that its mean is undefined because: — The integral of x*f(x) from -infinity to +infinity does not converge.
39. In a multinomial distribution with k possible outcomes, what do the parameters typically represent? — The number of trials and the probability of each outcome
40. The Negative Binomial distribution parameter 'r' typically represents: — The target number of successes.
41. The Gamma distribution is a continuous probability distribution often used to model: — The waiting time until a certain number of events occur
42. The Negative Binomial distribution describes the probability of: — The number of failures before the r-th success
43. The Negative Binomial distribution can be viewed as the distribution of the number of trials needed to achieve a fixed number of successes. What is the alternative definition? — The number of failures before the r-th success.
44. A common application of the Beta distribution is in Bayesian statistics for modeling: — The posterior distribution of a probability parameter.
45. In a binomial distribution B(n, p), what does 'n' represent? — The total number of independent trials
46. The sum of independent Gamma random variables with the same scale parameter is also a Gamma random variable. What is the shape parameter of the resulting distribution? — The sum of the original shape parameters.
47. The Beta distribution is often used to model probabilities or proportions, which are bounded between 0 and 1. If alpha > 1 and beta > 1, the Beta distribution will be: — Unimodal and concentrated around the mean
48. The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. It is often used to model: — Proportions or percentages
49. The variance of a Binomial distribution B(n, p) is always less than or equal to its mean (np). When is the variance equal to the mean? — When p = 0
50. The Hypergeometric distribution is used when sampling is done: — Without replacement
All One Line Questions
E-Book Probability distributions - binomial, Poisson, normal, gamma, beta, Cauchy, multinomial, hypergeometric, negative binomial Online Test Probability distributions - binomial, Poisson, normal, gamma, beta, Cauchy, multinomial, hypergeometric, negative binomial Question Bank Probability distributions - binomial, Poisson, normal, gamma, beta, Cauchy, multinomial, hypergeometric, negative binomial