1. What is the primary benefit of using random variables in probability theory?

A. They eliminate randomness B. They enable quantification of uncertainty C. They provide deterministic values D. They replace outcomes with probabilities

2. In a dataset with an outlier, which measure of central tendency is least affected?

A. Mean B. Mode C. Range D. Median

3. The mean of a dataset is best described as which of the following?

A. The most frequently occurring value B. The value exactly in the middle after sorting C. The sum of all values divided by the number of values D. The measure of variability in the dataset

4. Why is variance often preferred over the range as a measure of spread?

A. Variance only considers two values B. Range is affected more by outliers than variance C. Variance uses information from all data points D. Range is harder to calculate

5. The main difference between discrete and continuous random variables is:

A. Discrete variables can take an infinite number of values B. Continuous variables can only take whole numbers C. Discrete variables take countable outcomes, continuous variables take values from intervals D. Continuous variables only take positive values

6. Which property must a valid PMF always satisfy?

A. The sum of all probabilities must be zero B. Probabilities can be negative as long as they sum to one C. The sum of probabilities for all possible values equals one D. The PMF must be a continuous function

7. What does a Probability Mass Function (PMF) represent for a discrete random variable?

A. The cumulative probability up to a value B. The probability that the variable equals a specific value C. The variance of the variable D. The expected value of the variable

8. Which measure summarizes the central tendency by finding the value that divides the data into two equal halves?

A. Mean B. Mode C. Median D. Variance

9. What does a robust measure of central tendency means?

A. A measure unaffected significantly by extreme values B. The most complex measure C. The average value always D. The measure based on the range only

10. How is sample variance different from population variance?

A. Sample variance divides by n, population variance divides by n-1 B. Sample variance divides by n-1, population variance divides by n C. Sample variance is always smaller than population variance D. There is no difference between the two

11. What common mistake should be avoided when working with CDFs?

A. Confusing PMF/PDF with CDF B. Treating CDF as non-decreasing C. Using CDF for both discrete and continuous variables D. Summing probabilities less than or equal to x

12. In which of the following applications is the discrete binomial distribution especially useful?

A. Modelling continuous data B. Counting the number of successes in fixed trials with two outcomes C. Measuring time until failure in systems D. Calculating the cumulative sum of random variables

13. The mode of a dataset refers to:

A. The average value B. The value that appears most frequently C. The midpoint value D. The highest value

14. How is the CDF for a continuous random variable mathematically defined?

A. As the sum of all PMF values up to x B. As the derivative of the PDF at point x C. As the integral of the PDF from negative infinity to x D. As the product of PMF and PDF at x

15. Which of the following is NOT a use of the CDF?

A. Hypothesis testing B. Calculating probabilities involving ranges C. Finding non-cumulative probabilities directly D. Reliability and survival analysis

16. Which of the following describes a random variable?

A. A variable that takes fixed values only B. A function assigning numerical values to random outcomes C. A deterministic value given by an experiment D. The total probability of an event

17. What is indicated by a mode value in a dataset?

A. The center of the data distribution B. The point around which data is symmetric C. The most frequent data point in the set D. The average distance from the mean

18. What does variance measure in a dataset?

A. Central tendency B. The square of the mean C. Spread or variability of the data points D. The most frequent value

19. How is standard deviation related to variance?

A. It is the square of variance B. It is the square root of variance C. It is equal to variance multiplied by mean D. They are unrelated

20. In a fair six-sided die roll, what is the probability that the outcome is less than or equal to 3?

A. 1/2 B. 1/3 C. 1/6 D. 2/3

21. If a fair six-sided die is rolled, what is the PMF of any one face?

A. 1/3 B. 1/6 C. 1/2 D. 1/4

22. Why can the median be a more robust measure of central tendency than the mean?

A. It is easier to calculate B. It is affected strongly by outliers C. It is not distorted by extreme values D. It represents the mode and mean simultaneously

23. Which of the following is considered a measure of dispersion?

A. Mode B. Median C. Variance D. Expectation

24. What is a ‘random variable’?

A. A fixed value determined by chance B. A variable whose value is numerical outcome of a random process C. A constant D. The total set of all possible outcomes

25. Which of the following is NOT true about the CDF?

A. It applies to both discrete and continuous variables B. It gives probabilities for variables being less than or equal to a value C. It takes negative values for some ranges D. It fully characterizes the distribution

26. Why is variance an important measure in probability and statistics?

A. It measures the average value B. It indicates the frequency of the most common value C. It captures variability and consistency of the data D. It shows the middle value of the dataset

27. For a discrete random variable representing coin toss heads in two flips, what is the PMF of observing exactly one head?

A. 1/4 B. 1/2 C. 1/3 D. 3/4

28. How does the CDF behave as the random variable tends towards infinity?

A. It goes to zero B. It approaches 1 C. It oscillates D. It equals the mean of the distribution

29. How is the population variance \( \sigma^2 \) calculated?

A. Sum of squares of deviations divided by sample size minus one B. Sum of squares of deviations divided by population size C. Mean of values squared D. Sum of values divided by population size

30. In practice, why is the binomial distribution widely used?

A. Because it models continuous outcomes well B. Because it models countable outcomes with two possible results per trial C. Because it only requires the mean value for calculations D. Because it cannot represent probabilities of success

31. What does the population variance formula divide by?

A. n-1, the sample size minus one B. n, the population size C. n+1 D. The square root of n

32. For a discrete random variable, how is the CDF computed using the PMF?

A. By subtracting individual probabilities from 1 B. By integrating the PMF C. By summing all probabilities less than or equal to the given value D. By multiplying probabilities at the value

33. Which of the following statements about the CDF is true?

A. The CDF decreases as the value increases B. The CDF is always non-decreasing C. The CDF can be negative for some values D. The CDF is only defined for discrete variables

34. For a random variable representing heads obtained in two coin tosses, which of the following is true about its PMF values?

A. P(X=0) = 1/2 B. P(X=2) = 1/4 C. P(X=1) = 1/4 D. P(X=0) = 1/3

35. Which summary statistic best represents the "typical" value in a skewed dataset?

A. Mean B. Mode C. Median D. Variance

36. When calculating variance, what is the role of deviations from the mean?

A. They identify the central value of the dataset B. They measure how far each data point is from the mean C. They calculate the most likely outcome D. They sum all the data points

37. In a dataset: {2, 4, 5, 7, 100}, why does the mean not adequately represent central tendency?

A. Because the median is higher B. Because the dataset is too small C. Because the outlier 100 pulls the mean away from typical values D. Because standard deviation is zero

38. What is the main difference between the PMF and the CDF for a discrete random variable?

A. PMF gives cumulative probabilities, CDF gives exact probabilities B. PMF is used only for continuous variables, CDF for discrete variables C. PMF gives the probability of a specific value, CDF gives the probability of being less than or equal to a value D. PMF sums probabilities, CDF differentiates probabilities

39. What does the Cumulative Distribution Function (CDF) provide for any random variable?

A. The probability that the variable equals a specific value B. The probability that the variable is less than or equal to a given value C. The expected value of the variable D. The standard deviation of the variable

40. Which of the following is true about the Cumulative Distribution Function (CDF) of a random variable?

A. It decreases as the variable increases B. It gives the probability that the variable is greater than a value C. It provides the probability that the variable is less than or equal to a value D. It is always equal to zero for values less than the mean

41. Why is the median considered a robust measure of central tendency?

A. It is affected heavily by outliers B. It always equals the mean in symmetric distributions C. It is not influenced much by extreme values D. It gives highest frequency in the data

42. What does the expectation (mean) of a random variable quantify?

A. The most common outcome B. The long-run average value of the variable C. The variability of values D. The probability of the maximum value

43. A fair six-sided die is rolled once. What is the probability that the outcome is less than or equal to 4?

A. 1/6 B. 2/3 C. 1/2 D. 5/6

44. In a dataset where the mean height is 170 cm and standard deviation is 5 cm, which range best covers most individuals' heights?

A. Between 160 cm and 180 cm B. Between 165 cm and 175 cm C. Between 170 cm and 180 cm D. Between 175 cm and 185 cm

45. What does the Probability Mass Function (PMF) of a discrete random variable represent?

A. The total number of outcomes B. The probability that the variable takes a specific value C. The cumulative probability up to a certain value D. The mean value of the random variable

46. What does the mode of a dataset represent?

A. The average value B. The most common value C. The middle value when data is ordered D. The measure of spread in the dataset

47. If a discrete random variable's PMF at \(x=3\) is 0.2, what does this signify?

A. The random variable will never take 3 B. The cumulative probability up to 3 is 0.2 C. The random variable has a 20% chance of being equal to 3 D. The expected value is 3

48. What is the primary property of a CDF that must always hold?

A. It must be non-increasing B. It can decrease depending on the distribution C. It is always non-decreasing D. It is only defined for discrete variables

49. What characteristic differentiates a PMF from a probability density function (PDF)?

A. PMF is used for continuous variables, PDF for discrete B. PMF assigns probabilities to discrete values, PDF assigns densities to continuous variables C. PMF sums to less than 1, PDF sums to 1 D. PMF is always plotted as a curve, PDF as bars

50. Compared to population variance, how is the sample variance \( s^2 \) computed differently?

A. Dividing by n instead of n-1 B. Using mean + variance C. Dividing by n-1 instead of n D. Taking the square root of population variance