Continuous Random Variable MCQs with Solution - 1
Continuous distributions
Note: Correct Option is Marked (*)
1. Which of the following statements is not correct concerning the probability distribution of a continuous random variable?
a. the vertical coordinate is the probability density function
b. the range of the random variable is found on the x-axis
c. the total area represented under the curve will equal 1
*d. the area under the curve between points a and b represents the probability that X = a
e. the area under the curve represents the sum of probabilities for all possible outcomes
2. Which of the following is not a characteristic of the normal distribution?
a. it is a symmetrical distribution
*b. the mean is always zero
c. the mean, median and mode are equal
d. it is a bell-shaped distribution
e. the area under the curve equals one
3. Which of the following is not a correct statement?
a. the exponential distribution describes the Poisson process as a continuous random variable
b. the exponential distribution is a family of curves, which are completely described by the mean
*c. the mean of the exponential distribution is the inverse of the mean of the Poisson
d. the Poisson is a probability distribution for a discrete random variable while the exponential distribution is continuous
e. the area under the curve for an exponential distribution equals 1
4. Which of the following do the normal distribution and the exponential density function have in common?
a. both are bell-shaped
b. both are symmetrical distributions
c. both approach infinity as x approaches infinity
*d. both approach zero as x approaches infinity
e. all of the above are features common to both distributions
5. Which of the following statement is not true for an exponential distribution with parameter λ?
a. mean = 1 / λ
b. standard deviation = 1 / λ
c. the distribution is completely determined once the value of λ is known
d. the area under the curve is equal to one
*e. the distribution is a two-parameter distribution since the mean and standard deviation are equal
6. Which of the following distributions is suitable to model the length of time that elapses before the first employee passes through the security door of a company?
*a. exponential
b. normal
c. poisson
d. binomial
e. uniform
7. Which of the following distributions is suitable to measure the length of time that elapses between the arrival of cars at a petrol station pump?
a. normal
b. binomial
c. uniform
d. poisson
*e. exponential
8. A multiple-choice test has 30 questions. There are 4 choices for each question. A student who has not studied for the test decides to answer all the questions randomly by guessing the answer to each question. Which of the following probability distributions can be used to calculate the student’s chance of getting at least 20 questions right?
*a. Binomial distribution
b. Poisson distribution
c. Exponential distribution
d. Uniform distribution
e. Normal distribution
9. It is known that 20% of all vehicles parked on campus during the week do not have the required parking disk. A random sample of 10 cars is observed one Monday morning and X is the number in the sample that do not have the required parking disk. We can assume here that the probability distribution of X is:
*a. Binomial
b. Normal
c. Poisson
d. Exponential
e. Any continuous distribution will do
10. Which of the following statements is/are true regarding the normal distribution curve?
a. it is symmetrical
b. it is bell-shaped
c. it is asymptotic in that each end approaches the horizontal axis but never reaches it
d. its mean, median and mode are located at the same point
*e. all of the above statements are true
11. Indicate which of the statements below does not correctly apply to normal probability distributions:
a. they are all unimodal (ie: have a single mode)
b. they are all symmetrical
*c. they all have the same mean and standard deviation
d. the area under the probability curve is always equal to 1
e. for the standard normal distribution µ = 0 and σ = 1
12. Which of the following is not a characteristic of a binomial experiment?
a. there is a sequence of identical trials
*b. each trial results in two or more possible outcomes
c. the trials are independent of each other
d. the probability of success, p, is the same from one trial to another
e. all of the above are characteristics of a binomial experiment
13. Which probability distribution is appropriate for a count of events when the events of interest occur randomly, independently of one another and rarely?
a. normal distribution
b. exponential distribution
c. uniform distribution
*d. poisson distribution
e. binomial distribution
14. Which of the following cannot generate a Poisson distribution?
a. The number of cars arriving at a parking garage in a one-hour time interval
b. The number of telephone calls received in a ten-minute interval
*c. The number of customers arriving at a petrol station
d. The number of bacteria found in a cubic yard of soil
e. The number of misprints per page
15. The mean for the exponential distribution equals the mean for the Poisson distribution only when the former distribution has a mean equal to
*a. 1.0
b. 0.5
c. 0.25
d. 2.0
e. the means of the two distributions can never be equal
16. A larger standard deviation for a normal distribution with an unchanged mean indicates that the distribution becomes:
a. narrower and more peaked
*b. flatter and wider
c. more skewed to the right
d. more skewed to the left
e. a change in the standard deviation does not change the shape of the distribution
17. Which of the following statements regarding the probability density function, f(x), of the uniform distribution is correct?
a. the height of the density function differs for different values of X
b. the density function increases as the values of X increase
c. the density function is roughly “bell-shaped”
*d. the density function is constant for all values that X can assume
e. none of the above statements are true
18. Which of the following statements is correct?
a. The Exponential distribution is continuous and defined over the interval (-∞, ∞)
*b. The mean of the Poisson distribution (with parameter μ) equals the mean of the Exponential distribution (with parameter λ) only when μ = λ = 1
c. It is impossible for a Normal distribution to have a negative population mean
d. The Binomial distribution has equal mean and variance only when p = 0.5
e. The Uniform distribution is a discrete probability distribution
19. In a popular shopping centre, the waiting time for an ABSA ATM machine is found to be uniformly distributed between 1 and 5 minutes. What is the probability of waiting between 2 and 3 minutes to use the ATM?
*a. 0.25
b. 0.50
c. 0.75
d. 0.20
e. 0.40
20. In a popular shopping centre, the waiting time for an ABSA ATM machine is found to be uniformly distributed between 1 and 5 minutes. What is the probability of waiting between 2 and 4 minutes to use the ATM?
a. 0.25
*b. 0.50
c. 0.75
d. 0.20
e. 0.40
21. In a popular shopping centre, the waiting time for an ABSA ATM machine is found to be uniformly distributed between 1 and 5 minutes. What is the probability of waiting between 2 and 5 minutes to use the ATM?
a. 0.25
b. 0.50
*c. 0.75
d. 0.20
e. 0.40
22. In a popular shopping centre, the waiting time for an ABSA ATM machine is found to be uniformly distributed between 1 and 6 minutes. What is the probability of waiting between 2 and 3 minutes to use the ATM?
a. 0.25
b. 0.50
c. 0.75
*d. 0.20
e. 0.40
23. In a popular shopping centre, the waiting time for an ABSA ATM machine is found to be uniformly distributed between 1 and 6 minutes. What is the probability of waiting between 2 and 4 minutes to use the ATM?
a. 0.25
b. 0.50
c. 0.75
d. 0.20
*e. 0.40
24. In a popular shopping centre, the waiting time for an ABSA ATM machine is found to be uniformly distributed between 1 and 5 minutes. What is the probability of being able to use the ATM in the first 30 seconds of waiting?
a. 0.25
b. 0.50
c. 0.75
d. 0.20
*e. 0.125
25. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 3 hours. What is the probability that a patient will have to wait between 50 minutes and 2 hours to see a doctor?
*a. 0.500
b. 0.286
c. 0.643
d. 0.786
e. 0.714
26. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 3 hours. What is the probability that a patient will have to wait between 50 minutes and 1.5 hours to see a doctor?
a. 0.500
*b. 0.286
c. 0.643
d. 0.786
e. 0.714
27. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 3 hours. What is the probability that a patient will have to wait between 30 minutes and 2 hours to see a doctor?
a. 0.500
b. 0.286
*c. 0.643
d. 0.786
e. 0.714
28. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 3 hours. What is the probability that a patient will have to wait between 40 minutes and 2.5 hours to see a doctor?
a. 0.500
b. 0.286
c. 0.643
*d. 0.786
e. 0.714
29. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 3 hours. What is the probability that a patient will have to wait between 50 minutes and 2.5 hours to see a doctor?
a. 0.500
b. 0.286
c. 0.643
d. 0.786
*e. 0.714
30. A train arrives at a station every 20 minutes. What is the probability that a person arriving at the station will have to wait less than half an hour for the next train?
a. 0.524
b. 0.237
c. 0.500
*d. 1.000
e. 0.872
31. In a small town the town hall clock strikes every half an hour. If you wake up at random during the middle of the night, what is the probability that you will have to wait less than 5 minutes before hearing the clock strike again?
a. 0.500
*b. 0.167
c. 0.833
d. 0.457
e. 0.138
32. It is known that the amount of apple juice found in 500ml bottle is uniformly distributed between 495ml and 510ml. What is the probability that a randomly selected bottle of apple juice contains less than 500ml of juice?
*a. 0.333
b. 0.667
c. 0.500
d. 0.000
e. 1.000
33. It is known that the amount of apple juice found in 500ml bottle is uniformly distributed between 495ml and 510ml. What is the probability that a randomly selected bottle of apple juice contains more than 500ml of juice?
a. 0.333
*b. 0.667
c. 0.500
d. 0.000
e. 1.000
34. The mass of a 1000g container of yoghurt is equally likely to take on any value in the interval (995g,1010g). The container will not contain less than 995g or more than 1010g of yoghurt. What is the probability that a randomly chosen container of yoghurt has a mass of less than 1000g?
*a. 0.333
b. 0.667
c. 0.500
d. 0.000
e. 1.000
35. An investor knows that his portfolio is equally likely to yield an annual return anywhere in the interval [5%, 35%]. What is the probability that he will earn more than 13.5% in the forthcoming year?
*a. 0.72
b. 0.50
c. 0.42
d. 0.17
e. 0.83
36. An investor knows that his portfolio is equally likely to yield an annual return anywhere in the interval [5%, 35%]. What is the probability that he will earn more than 20% in the forthcoming year?
a. 0.72
*b. 0.50
c. 0.42
d. 0.17
e. 0.83
37. An investor knows that his portfolio is equally likely to yield an annual return anywhere in the interval [5%, 35%]. What is the probability that he will earn more than 22.5% in the forthcoming year?
a. 0.72
b. 0.50
*c. 0.42
d. 0.17
e. 0.83
38. An investor knows that his portfolio is equally likely to yield an annual return anywhere in the interval [5%, 35%]. What is the probability that he will earn more than 30% in the forthcoming year?
a. 0.72
b. 0.50
c. 0.42
*d. 0.17
e. 0.83
39. An investor knows that his portfolio is equally likely to yield an annual return anywhere in the interval [5%, 35%]. What is the probability that he will earn more than 10% in the forthcoming year?
a. 0.72
b. 0.50
c. 0.42
d. 0.17
*e. 0.83
40. The distance between Cape Town Cape Town ) and you have been dropped off at some point along the highway between Cape Town
a. 0.625
b. 0.375
c. 0.750
d. 0.666
*e. 0.500
41. Everyday you make the trip from your home to work by car. The travel time to work from home is equally likely to be anywhere between 10 minutes and 25 minutes, but is never less than 10 minutes or more than 25 minutes. What is the probability that on a randomly selected day it takes you more than 20 minutes to travel to work from home?
a. 1.000
*b. 0.333
c. 0.500
d. 0.667
e. 0.000
42. Everyday you make the trip from your home to work by car. The travel time to work from home is equally likely to be anywhere between 10 minutes and 25 minutes, but is never less than 10 minutes or more than 25 minutes. What is the probability that on a randomly selected day it takes you more than 15 minutes to travel to work from home?
a. 1.000
b. 0.333
c. 0.500
*d. 0.667
e. 0.000
43. The temperature in your office at work is regulated by a thermostat and is equally likely to be between 17 degrees Celsius and 23 degrees Celsius, but is never more than 23 or less than 17 degrees. What is the probability that at a randomly selected point in time, the temperature in your office is more than 21 degrees?
*a. 0.333
b. 0.500
c. 0.667
d. 0.015
e. 0.833
44. The temperature in your office at work is regulated by a thermostat and is equally likely to be between 17 degrees Celsius and 23 degrees Celsius, but is never more than 23 or less than 17 degrees. What is the probability that at a randomly selected point in time, the temperature in your office is more than 18 degrees?
a. 0.333
b. 0.500
c. 0.667
d. 0.015
*e. 0.833
45. The temperature in your office at work is regulated by a thermostat and is equally likely to be between 17 degrees Celsius and 23 degrees Celsius, but is never more than 23 or less than 17 degrees. What is the probability that at a randomly selected point in time, the temperature in your office is less than 21 degrees?
a. 0.333
b. 0.500
*c. 0.667
d. 0.015
e. 0.833
46. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 3 hours. What is the expected waiting time for patients to see a doctor?
*a. 110 minutes
b. 80 minutes
c. 95 minutes
d. 60 minutes
e. 100 minutes
47. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 2 hours. What is the expected waiting time for patients to see a doctor?
a. 110 minutes
*b. 80 minutes
c. 95 minutes
d. 60 minutes
e. 100 minutes
48. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 2.5 hours. What is the expected waiting time for patients to see a doctor?
a. 110 minutes
b. 80 minutes
*c. 95 minutes
d. 60 minutes
e. 100 minutes
49. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 30 minutes and 1.5 hours. What is the expected waiting time for patients to see a doctor?
a. 110 minutes
b. 80 minutes
c. 95 minutes
*d. 60 minutes
e. 100 minutes
50. The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 50 minutes and 2.5 hours. What is the expected waiting time for patients to see a doctor?
a. 110 minutes
b. 80 minutes
c. 95 minutes
d. 60 minutes
*e. 100 minutes
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