TRB PG Assistant - Probability distributions - binomial, Poisson, normal, gamma, beta, Cauchy, multinomial, hypergeometric, negative binomial Online Test (MCQ Quiz)

1. 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?

A) Poisson distribution B) Normal distribution C) Binomial distribution D) Gamma distribution

2. In a binomial distribution B(n, p), what does 'n' represent?

A) The probability of success in a single trial B) The number of failures C) The mean of the distribution D) The total number of independent trials

3. For a binomial distribution B(n, p), what is the variance?

A) np B) np(1-p) C) sqrt(np(1-p)) D) n(1-p)

4. 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?

A) n (number of trials) B) p (probability of success) C) lambda (average rate of occurrence) D) sigma (standard deviation)

5. If X follows a Poisson distribution with parameter lambda, what is the expected value (mean) of X?

A) lambda^2 B) lambda C) 1/lambda D) sqrt(lambda)

6. The variance of a Poisson distribution with parameter lambda is:

A) lambda B) lambda^2 C) sqrt(lambda) D) 1

7. Which continuous probability distribution is bell-shaped, symmetric, and characterized by its mean (mu) and standard deviation (sigma)?

A) Gamma distribution B) Beta distribution C) Cauchy distribution D) Normal distribution

8. In a standard normal distribution, what are the values of the mean and standard deviation?

A) Mean = 1, Standard Deviation = 0 B) Mean = 0, Standard Deviation = 1 C) Mean = 0, Standard Deviation = 0 D) Mean = 1, Standard Deviation = 1

9. The Gamma distribution is a continuous probability distribution often used to model:

A) The number of successes in a fixed number of trials B) The waiting time until a certain number of events occur C) The outcome of a single event with two possibilities D) The distribution of sample means

10. What are the parameters of a Gamma distribution, typically denoted by k (shape) and theta (scale)?

A) Mean and Variance B) Probability of success and number of trials C) Shape and Scale D) Location and Scale

11. The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. It is often used to model:

A) Waiting times B) Proportions or percentages C) Counts of rare events D) Any non-negative random variable

12. What are the parameters of a Beta distribution, typically denoted by alpha and beta?

A) Location and Scale B) Shape and Scale C) Rate and Count D) Two positive shape parameters

13. The Cauchy distribution is known for its heavy tails and lack of a well-defined mean or variance. What is its characteristic parameter?

A) Lambda (rate) B) Mu (location) and Sigma (scale) C) k (shape) and theta (scale) D) n (trials) and p (probability)

14. The Multinomial distribution is a generalization of the binomial distribution. It describes the outcome of:

A) A single trial with two possible outcomes B) Multiple independent trials, each with two outcomes C) A fixed number of independent trials, each with more than two possible outcomes D) The number of events in a fixed interval

15. In a multinomial distribution with k possible outcomes, what do the parameters typically represent?

A) The mean and variance for each outcome B) The number of trials and the probability of each outcome C) The number of successes and failures D) The location and scale parameters

16. The Hypergeometric distribution is used when sampling is done:

A) With replacement B) Without replacement C) From an infinite population D) From a normally distributed population

17. Which parameters define a Hypergeometric distribution?

A) n (number of trials) and p (probability of success) B) lambda (average rate) C) N (population size), K (number of success states in population), and n (number of draws) D) mu (mean) and sigma (standard deviation)

18. The Negative Binomial distribution describes the probability of:

A) The number of successes in a fixed number of trials B) The number of failures before the r-th success C) The waiting time for the first event D) The outcome of a single trial with multiple possibilities

19. What is a common parameterization for the Negative Binomial distribution, often denoted by r and p?

A) r = number of successes, p = probability of success B) r = number of failures, p = probability of success C) r = number of trials, p = probability of failure D) r = number of successes, p = probability of failure

20. 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?

A) Normal distribution B) Gamma distribution C) Poisson distribution D) Beta distribution

21. 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?

A) np > 5 and n(1-p) > 5 B) np < 5 and n(1-p) < 5 C) np = 5 D) p = 0.5

22. Which of the following is a property of the Normal distribution?

A) It is always skewed to the right. B) It has a fixed mean and standard deviation. C) It is symmetric about its mean. D) It is defined only for positive values.

23. The Gamma function, denoted by Gamma(z), is fundamental to the Gamma distribution. What is Gamma(n) for a positive integer n?

A) n! B) (n-1)! C) n D) 1/n!

24. In the context of the Beta distribution, if alpha = 1 and beta = 1, what distribution does it represent?

A) A degenerate distribution at 0 B) A uniform distribution on [0, 1] C) A normal distribution D) A Poisson distribution

25. The Cauchy distribution is sometimes called the 'Lorentz distribution'. A key characteristic is that its mean is undefined because:

A) The integral of x*f(x) from -infinity to +infinity does not converge. B) The distribution is only defined for positive values. C) The distribution has a single peak. D) The variance is zero.

26. A single draw from a Multinomial distribution with parameters n and p1, p2, ..., pk results in:

A) A single value representing one of the k outcomes. B) A vector of k counts, summing to n, representing the number of times each outcome occurred. C) A continuous value between 0 and 1. D) A value representing the number of trials.

27. 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?

A) Binomial distribution B) Poisson distribution C) Hypergeometric distribution D) Negative Binomial distribution

28. 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?

A) The number of successes in a fixed number of trials. B) The number of failures before the r-th success. C) The time until the first event. D) The sum of two independent exponential random variables.

29. Which distribution is characterized by the probability density function f(x) = (1 / (sigma * sqrt(2*pi))) * exp(-0.5 * ((x - mu) / sigma)^2)?

A) Gamma distribution B) Cauchy distribution C) Normal distribution D) Beta distribution

30. 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?

A) The sum of the original shape parameters. B) The product of the original shape parameters. C) The average of the original shape parameters. D) The scale parameter.

31. A common application of the Beta distribution is in Bayesian statistics for modeling:

A) The prior distribution of a count variable. B) The posterior distribution of a probability parameter. C) The likelihood function of a normal variable. D) The distribution of a variance.

32. The Cauchy distribution has a characteristic function (phi(t)) given by exp(|t|). This implies:

A) It has a finite mean. B) It has a finite variance. C) It is symmetric about the origin. D) It has heavy tails.

33. 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?

A) Binomial distribution B) Multinomial distribution C) Hypergeometric distribution D) Poisson distribution

34. 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:

A) Binomial distribution B) Poisson distribution C) Hypergeometric distribution D) Negative Binomial distribution

35. Consider a sequence of Bernoulli trials. The Negative Binomial distribution models the probability of obtaining:

A) A specific number of successes in a fixed number of trials. B) A specific number of failures before the k-th success. C) The number of successes until the first failure. D) The total number of trials until the r-th failure.

36. 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)?

A) Normal distribution B) Beta distribution C) Gamma distribution D) Exponential distribution

37. 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:

A) Calculating moments of the Beta distribution. B) Deriving the PDF of the Beta distribution. C) Both A and B. D) Neither A nor B.

38. The median of a standard Normal distribution (mu=0, sigma=1) is:

A) 0 B) 1 C) 0.5 D) Undefined

39. 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:

A) lambda is large. B) k is small. C) lambda > 0. D) k is an integer.

40. In a binomial experiment, if 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) 0.3125

41. Which of the following statements about the Normal distribution is FALSE?

A) The curve is bell-shaped. B) The curve is symmetric about the mean. C) The mean, median, and mode are equal. D) The total area under the curve is infinite.

42. The Gamma distribution is a generalization of the exponential distribution. If the shape parameter k=1, the Gamma distribution becomes:

A) A Normal distribution B) An Exponential distribution C) A Beta distribution D) A Chi-squared distribution

43. The Cauchy distribution's heavy tails mean that:

A) Extreme values are more likely than in a normal distribution. B) Extreme values are less likely than in a normal distribution. C) The distribution is concentrated around the mean. D) The variance is very small.

44. The sum of 'n' independent and identically distributed standard normal random variables follows a:

A) Normal distribution with mean n and variance n B) Chi-squared distribution with n degrees of freedom C) Gamma distribution with shape n and scale 1 D) T-distribution with n degrees of freedom

45. 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:

A) U-shaped B) Bell-shaped C) Unimodal and concentrated around the mean D) Bimodal

46. 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?

A) Binomial B) Poisson C) Hypergeometric D) Multinomial

47. The Negative Binomial distribution parameter 'r' typically represents:

A) The number of failures. B) The number of trials. C) The target number of successes. D) The probability of success.

48. The limiting distribution of the sum of a large number of independent and identically distributed random variables (under certain conditions) is the:

A) Poisson distribution B) Binomial distribution C) Normal distribution D) Gamma distribution

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?

A) When p = 1 B) When p = 0 C) When p = 0.5 D) Never

50. 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:

A) Mean and Variance B) Shape and Scale C) Location and Scale D) Number of trials and Probability of success

Probability distributions - binomial, Poisson, normal, gamma, beta, Cauchy, multinomial, hypergeometric, negative binomial - Online Test

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