Probability distributions - binomial, Poisson, normal, gamma, beta, Cauchy, multinomial, hypergeometric, negative binomial - Question Bank
1. 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:
2. 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?
3. The limiting distribution of the sum of a large number of independent and identically distributed random variables (under certain conditions) is the:
4. The Negative Binomial distribution parameter 'r' typically represents:
5. 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?
6. 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:
7. The sum of 'n' independent and identically distributed standard normal random variables follows a:
8. The Cauchy distribution's heavy tails mean that:
9. The Gamma distribution is a generalization of the exponential distribution. If the shape parameter k=1, the Gamma distribution becomes:
10. Which of the following statements about the Normal distribution is FALSE?
11. In a binomial experiment, if n=10 and p=0.5, what is the probability of getting exactly 5 successes?
12. 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:
13. The median of a standard Normal distribution (mu=0, sigma=1) is:
14. 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:
15. 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)?
16. Consider a sequence of Bernoulli trials. The Negative Binomial distribution models the probability of obtaining:
17. 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:
18. 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?
19. The Cauchy distribution has a characteristic function (phi(t)) given by exp(|t|). This implies:
20. A common application of the Beta distribution is in Bayesian statistics for modeling:
21. 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?
22. Which distribution is characterized by the probability density function f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-0.5 * ((x - mu) / sigma)^2)?
23. 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?
24. 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?
25. A single draw from a Multinomial distribution with parameters n and p1, p2, ..., pk results in:
26. The Cauchy distribution is sometimes called the 'Lorentz distribution'. A key characteristic is that its mean is undefined because:
27. In the context of the Beta distribution, if alpha = 1 and beta = 1, what distribution does it represent?
28. The Gamma function, denoted by Gamma(z), is fundamental to the Gamma distribution. What is Gamma(n) for a positive integer n?
29. Which of the following is a property of the Normal distribution?
30. 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?
31. 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?
32. What is a common parameterization for the Negative Binomial distribution, often denoted by r and p?
33. The Negative Binomial distribution describes the probability of:
34. Which parameters define a Hypergeometric distribution?
35. The Hypergeometric distribution is used when sampling is done:
36. In a multinomial distribution with k possible outcomes, what do the parameters typically represent?
37. The Multinomial distribution is a generalization of the binomial distribution. It describes the outcome of:
38. The Cauchy distribution is known for its heavy tails and lack of a well-defined mean or variance. What is its characteristic parameter?
39. What are the parameters of a Beta distribution, typically denoted by alpha and beta?
40. The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. It is often used to model:
41. What are the parameters of a Gamma distribution, typically denoted by k (shape) and theta (scale)?
42. The Gamma distribution is a continuous probability distribution often used to model:
43. In a standard normal distribution, what are the values of the mean and standard deviation?
44. Which continuous probability distribution is bell-shaped, symmetric, and characterized by its mean (mu) and standard deviation (sigma)?
45. The variance of a Poisson distribution with parameter lambda is:
46. If X follows a Poisson distribution with parameter lambda, what is the expected value (mean) of X?
47. 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?
48. For a binomial distribution B(n, p), what is the variance?
49. In a binomial distribution B(n, p), what does 'n' represent?
50. 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?