1. What defines an independent event in probability?

A. Events that cannot happen at the same time B. Events whose occurrence does not affect one another C. Events that have the same probability D. Events that occur simultaneously always

2. The Poisson distribution models:

A. Number of successes in fixed trials B. Events occurring in a fixed interval of time or space C. Failures before first success D. Number of trials to a fixed number of successes

3. Which of the following is not a property of the normal distribution?

A. It is bell-shaped B. Mean = median = mode C. It is skewed to the right D. The area under the curve totals 1

4. For a normal random variable \( X \) with mean \( \mu \) and standard deviation \( \sigma \), the standardized variable \( Z \) is:

A. \( Z = \frac{X - \sigma}{\mu} \) B. \( Z = \frac{X - \mu}{\sigma} \) C. \( Z = X + \mu \) D. \( Z = \frac{X}{\sigma} \)

5. Which of the following best describes a continuous random variable?

A. It can only take integer values B. It takes any value within a continuous interval C. Its values are limited to a finite set D. It is used only when outcomes are categorical

6. If X follows a Negative Binomial distribution with parameters r and p, which situation is modeled?

A. Number of successes before the first failure B. Number of failures before r successes C. Number of events in a fixed interval D. Number of successes in n trials

7. The mean of a hypergeometric distribution with parameters N, K, n is:

A. nK/N B. KN/n C. NK/n D. N/n

8. The mean of a Geometric distribution is:

A. \( 1-p \) B. \( \frac{1}{p} \) C. \( \frac{p}{1-p} \) D. \( p \)

9. In probability, the axiom stating that the probability of the sample space S is 1 means:

A. At least one event is certain to occur B. No events can occur C. All events are equally likely D. Events are all mutually exclusive

10. The Normal distribution is symmetric, so:

A. Mean > Median > Mode B. Mean < Median < Mode C. Mean = Median = Mode D. Mean and Mode are equal, but median differs

11. The Geometric distribution models:

A. Number of successes before a fixed number of failures B. Number of failures before the first success C. Number of successes in fixed number of trials D. Number of failures after fixed number of successes

12. What does the variance V(X) measure in a random variable?

A. The average value of X B. The spread or dispersion of the values of X about the mean C. The probability of success in a trial D. The median of the distribution

13. In the binomial example about luncheon vouchers with error rate 7%, what is the probability of exactly two vouchers having errors?

A. 0.0049 B. 0.03941 C. 0.804357 D. 0.26182

14. Which distribution is used to model the number of successes in \( n \) independent Bernoulli trials?

A. Poisson distribution B. Binomial distribution C. Normal distribution D. Geometric distribution

15. For a Negative Binomial distribution \( (r, p) \), the mean is:

A. \( \frac{r}{p} \) B. \( r p \) C. \( r \frac{1-p}{p} \) D. \( \frac{p}{r} \)

16. In a normal distribution with mean μ and standard deviation σ, which of the following statements is TRUE regarding the standardized variable Z?

A. Z has the same mean and standard deviation as X B. Z has mean 0 and standard deviation 1 C. Z can only take positive values D. Z represents the raw score in the distribution

17. The probability mass function of a Geometric distribution is:

A. \( P(X=x) = p(1-p)^x \) for \( x=0,1,2,\ldots \) B. \( P(X=x) = (1-p)^x p \) for \( x=0,1,2,\ldots \) C. \( P(X=x) = \binom{n}{x} p^x (1-p)^{n-x} \) D. \( P(X=x) = \frac{e^{-\lambda} \lambda^x}{x!} \)

18. Which of the following defines mutually exclusive events?

A. Events that can happen at the same time B. Events whose outcomes are dependent C. Events that cannot happen together D. Events with equal probabilities

19. In the same example, what is the probability that a farmer produces more than 780 kg of cocoa?

A. 0.0606 B. 0.3102 C. 0.5 D. 0.46

20. In the normal distribution example, what is the probability that a farmer produces less than 650 kg of cocoa?

A. 0.9394 B. 0.0606 C. 0.3102 D. 0.46

21. What is the probability mass function (pmf) of a Bernoulli distribution with parameter \( p \)?

A. \( P(X=1) = p, P(X=0) = 1-p \) B. \( P(X=1) = 1-p, P(X=0) = p \) C. \( P(X=x) = \frac{1}{2} \) for \( x=0,1 \) D. \( P(X=x) = px \)

22. The variance of a binomial distribution is given by:

A. np B. npq C. n(1-p) D. p(1-p)

23. The hypergeometric distribution applies when:

A. Sampling is done with replacement B. Sampling is done without replacement C. Trials are independent D. Events are mutually exclusive

24. What is the mean of a binomial distribution with parameters n and p?

A. n/p B. npq C. np D. p/n

25. The mean of a Bernoulli distribution with parameter \( p \) is:

A. \( p(1-p) \) B. \( 1-p \) C. \( p \) D. \( \sqrt{p} \)

26. What is the mean (expected value) \( E(X) \) of a discrete random variable \( X \)?

A. \( E(X) = \sum_i p_i \) B. \( E(X) = \sum_i x_i p_i \) C. \( E(X) = \max(x_i) \) D. \( E(X) = \min(x_i) \)

27. In the Normal distribution, within 1 standard deviation from the mean, the approximate area under the curve is:

A. 68.27% B. 95.45% C. 99.73% D. 50%

28. The total probability of all outcomes in a sample space \( S \) equals:

A. 0 B. 0.5 C. 1 D. Depends on experiments

29. The binomial distribution describes:

A. The number of successes in a fixed number of independent trials B. The time until the first success occurs C. The number of failures before the first success D. The sum of two independent Poisson variables

30. The Negative Binomial distribution generalizes the Geometric distribution by:

A. Counting failures before first success B. Counting failures before \( r \) successes C. Counting successes in \( r \) trials D. Counting successes before first failure

31. For a normal distribution with mean 750 kg and standard deviation 65 kg, what does a Z-score of -1.54 for X=650 indicate?

A. X is 1.54 standard deviations above the mean B. X is 1.54 standard deviations below the mean C. X is the mean value D. X is 650 standard deviations from the mean

32. What does an event being mutually exclusive mean?

A. It must occur together with another event B. It cannot occur simultaneously with certain other events C. It has probability zero D. It happens always

33. The Poisson distribution is used primarily to model:

A. The number of successes in independent trials B. The number of failures before a success C. The number of events occurring in a fixed interval D. The time to failure in a process

34. How is variance \( V(X) \) of a random variable defined?

A. \( V(X) = E(X)^2 \) B. \( V(X) = E(X^2) - [E(X)]^2 \) C. \( V(X) = E(X^2) + [E(X)]^2 \) D. \( V(X) = E(X^2) \)

35. The mean of the Hypergeometric distribution with parameters \( N, K, n \) is:

A. \( \frac{nk}{N} \) B. \( n p \) C. \( \frac{n}{N} \) D. \( \frac{K}{n} \)

36. What is the definition of probability for an event \( A \) in a sample space \( S \)?

A. The ratio of favorable outcomes of \( A \) to all possible outcomes in \( S \) B. The total number of outcomes in \( S \) C. The number of events opposite to \( A \) D. The sum of outcomes in \( S \)

37. What is the pmf of a Poisson distribution with mean \( \lambda \)?

A. \( P(X=x) = \frac{\lambda^x e^{-\lambda}}{x!} \) B. \( P(X=x) = \binom{n}{x} p^x (1-p)^{n-x} \) C. \( P(X=x) = p(1-p)^x \) D. \( P(X=x) = \frac{1}{2} \)

38. The mean and variance of a Poisson distribution with parameter \( \lambda \) are:

A. Mean \( = \lambda \), Variance \( = \lambda^2 \) B. Mean \( = \lambda \), Variance \( = \lambda \) C. Mean \( = \lambda^2 \), Variance \( = \lambda \) D. Mean \( = \sqrt{\lambda} \), Variance \( = \lambda \)

39. What is \( P(Z > 0.46) \) approximately?

A. 0.6772 B. 0.3102 C. 0.1500 D. 0.5400

40. The Hypergeometric distribution applies when sampling:

A. With replacement from an infinite population B. Without replacement from a finite population C. Using independent Bernoulli trials D. When events follow Poisson process

41. The standard normal distribution has:

A. Mean 1 and standard deviation 0 B. Mean 0 and standard deviation 1 C. Mean 0 and variance 0 D. Mean 1 and variance 1

42. The variance of a Binomial distribution \( B(n,p) \) is:

A. \( npq \) where \( q=1-p \) B. \( np \) C. \( n^2 p \) D. \( np^2 \)

43. What is the fundamental difference between a Geometric and a Negative Binomial distribution?

A. Geometric models number of failures before first success, Negative Binomial before r successes B. Negative Binomial models first success only C. Geometric assumes dependent trials D. There is no difference; both model the same situation

44. The binomial coefficient \( \binom{n}{x} \) is expressed as:

A. \( \frac{n!}{x! (n-x)!} \) B. \( \frac{x!}{n!} \) C. \( \frac{n!}{(n+x)!} \) D. \( \frac{(n-x)!}{n!} \)

45. What characterizes a discrete random variable?

A. It can take any value within an interval B. It takes values from an uncountably infinite set C. It assumes a countable number of distinct values D. It always takes whole numbers greater than 100

46. What is the mean \( E(X) \) of a Binomial distribution \( B(n,p) \)?

A. \( npq \) B. \( np \) C. \( n+p \) D. \( \sqrt{npq} \)

47. Why is it convenient to use the standardized variable Z instead of X in normal distribution calculations?

A. Because Z always equals X minus standard deviation B. Because Z always has mean 0 and standard deviation 1 C. Because X is always larger than Z D. Because Z is measured in the original units of X

48. Approximately what percentage of the area under the normal curve lies within 2 standard deviations from the mean?

A. 50% B. 68.27% C. 95.45% D. 99.73%

49. What is \( P(Z < -1.54) \) approximately using the standard normal distribution?

A. 0.0606 B. 0.1540 C. 0.3098 D. 0.8700

50. When is the binomial distribution used?

A. When trials are dependent B. When each trial has more than two outcomes C. When there are independent trials each with two possible outcomes D. When events are mutually exclusive only