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Unit 6: Probability and Stochastic Processes - Practice Quiz

ECE183 — Mathematics For Robotics 60 Questions
0 Correct 0 Wrong 60 Left
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1 Bayes' theorem is mainly used to calculate which probability?

Bayes' theorem Easy
A. A sample size
B. A total probability
C. A conditional probability
D. A prior probability

2 If , , and , what is ?

Bayes' theorem Easy
A.
B.
C.
D.

3 In Bayes' theorem, what does usually represent?

Bayes' theorem Easy
A. The conditional evidence
B. The prior probability
C. The posterior probability
D. The joint probability

4 A probability mass function (PMF) is used for which type of random variable?

Probability distributions and probability mass functions Easy
A. A constant random variable
B. A negative random variable
C. A discrete random variable
D. A continuous random variable

5 What is the sum of all probabilities in a valid probability mass function?

Probability distributions and probability mass functions Easy
A.
B.
C.
D.

6 If a discrete random variable has and , what is ?

Probability distributions and probability mass functions Easy
A.
B.
C.
D.

7 What does the cumulative distribution function represent?

Cumulative distribution functions Easy
A.
B.
C.
D.

8 A cumulative distribution function always has values between which limits?

Cumulative distribution functions Easy
A. and
B. and
C. and
D. and

9 As increases, a cumulative distribution function can only do what?

Cumulative distribution functions Easy
A. Stay the same or increase
B. Remain negative
C. Decrease
D. Become greater than

10 A probability density function (PDF) is generally used for which type of random variable?

Probability density functions Easy
A. A discrete random variable
B. A variable with one possible value
C. A continuous random variable
D. A Boolean variable only

11 For a continuous random variable, what is the probability of one exact value, such as ?

Probability density functions Easy
A.
B.
C.
D.

12 The total area under a valid probability density function is equal to what?

Probability density functions Easy
A.
B.
C.
D.

13 What does the mean of a probability distribution describe?

Mean and variance Easy
A. The smallest possible value
B. The largest possible value
C. The number of outcomes
D. The typical or average value

14 What does variance measure in a probability distribution?

Mean and variance Easy
A. The center of the data
B. The probability of one event
C. The total number of trials
D. The spread around the mean

15 If every value in a dataset is identical, what is its variance?

Mean and variance Easy
A.
B. It cannot be calculated
C.
D.

16 Which situation is modeled by a binomial distribution?

Binomial distribution Easy
A. The exact time between random arrivals
B. A measurement with infinitely many values
C. A fixed number of independent yes-or-no trials
D. A process with changing success probabilities

17 In a binomial distribution, what does usually represent?

Binomial distribution Easy
A. The success probability
B. The number of successes
C. The mean value
D. The number of trials

18 For a binomial random variable with trials, what is the maximum possible number of successes?

Binomial distribution Easy
A.
B.
C.
D.

19 The Poisson distribution is commonly used to model what?

Poisson distribution Easy
A. The number of events in a fixed interval
B. The outcome of one coin toss
C. The average of two measurements
D. The maximum value in a dataset

20 What parameter is commonly used in a Poisson distribution?

Poisson distribution Easy
A. , the standard deviation only
B. , the number of fixed trials
C. , the average event rate
D. , the probability of two outcomes

21 A robot uses a sensor to detect a surface defect. Defects occur on of surfaces. The sensor correctly detects a defect with probability and incorrectly signals a defect on a normal surface with probability . Given that the sensor signals a defect, what is the probability that a defect is actually present?

Bayes' theorem Medium
A.
B.
C.
D.

22 A robot encounters an obstacle with probability . Its sensor detects the obstacle with probability when one is present and gives a false alarm with probability when none is present. What is the probability that an obstacle is present given that the sensor detects one?

Bayes' theorem Medium
A.
B.
C.
D.

23 A discrete random variable has probability mass function , , and . What is ?

Probability distributions and probability mass functions Medium
A.
B.
C.
D.

24 A discrete random variable has pmf for . What value of makes this a valid probability mass function?

Probability distributions and probability mass functions Medium
A.
B.
C.
D.

25 For a random variable , and . What is ?

Cumulative distribution functions Medium
A.
B.
C.
D.

26 A random variable has support with , , and . What is ?

Cumulative distribution functions Medium
A.
B.
C.
D.

27 A continuous random variable has pdf for and otherwise. What is ?

Probability density functions Medium
A.
B.
C.
D.

28 The position error of a robot is uniformly distributed on . What is the probability that ?

Probability density functions Medium
A.
B.
C.
D.

29 A random variable takes values , , and with probabilities , , and , respectively. What is the variance of ?

Mean and variance Medium
A.
B.
C.
D.

30 A measurement error has mean and variance . If a new variable is defined by , what are the mean and variance of ?

Mean and variance Medium
A. Mean , variance
B. Mean , variance
C. Mean , variance
D. Mean , variance

31 Independent random variables and have means and and variances and , respectively. For , what are and ?

Mean and variance Medium
A. ,
B. ,
C. ,
D. ,

32 A quality-control robot tests components, each having an independent probability of being defective. What is the probability that exactly components are defective?

Binomial distribution Medium
A.
B.
C.
D.

33 A robot successfully completes a maneuver with probability on each independent attempt. In attempts, what is the probability of at least one failure?

Binomial distribution Medium
A.
B.
C.
D.

34 The number of successful grasps in independent trials follows a binomial distribution with success probability . What are its mean and variance?

Binomial distribution Medium
A. Mean , variance
B. Mean , variance
C. Mean , variance
D. Mean , variance

35 A robot receives an average of wireless commands per minute. Assuming a Poisson model, what is the probability of receiving exactly commands in one minute?

Poisson distribution Medium
A.
B.
C.
D.

36 A sensor reports faults at an average rate of per hour. Assuming a Poisson process, what is the probability of no faults during a -hour interval?

Poisson distribution Medium
A.
B.
C.
D.

37 The number of communication errors in a data packet follows a Poisson distribution with parameter . What are the mean and variance of the number of errors?

Poisson distribution Medium
A. Mean , variance
B. Mean , variance
C. Mean , variance
D. Mean , variance

38 A robot's positioning error is normally distributed with mean mm and standard deviation mm. What is the standardized score for an error of mm?

Normal distribution Medium
A.
B.
C.
D.

39 The completion time for a robotic task is normally distributed with mean seconds and standard deviation seconds. Approximately what proportion of tasks take between and seconds?

Normal distribution Medium
A.
B.
C.
D.

40 A robot's battery life is normally distributed with mean minutes and standard deviation minutes. What is the approximate probability that the battery lasts more than minutes?

Normal distribution Medium
A.
B.
C.
D.

41 A robot's obstacle detector has sensitivity and false-positive rate . Obstacles occur independently at a prior probability of . Given a positive detection, what is the probability that an obstacle is actually present?

Bayes' theorem Hard
A.
B.
C.
D.

42 A robot classifies terrain as type with prior probability . Two independent sensor readings are obtained conditional on the terrain type. Their likelihoods are , , , and . What is ?

Bayes' theorem Hard
A.
B.
C.
D.

43 A discrete random variable has probability mass function for . Which statement is correct?

Probability distributions and probability mass functions Hard
A. and the mean is infinite
B. and the mean is finite but the variance is infinite
C. and the variance is finite
D. and both the mean and variance are finite

44 Suppose has probability mass function for . What is ?

Probability distributions and probability mass functions Hard
A.
B.
C.
D.

45 A random variable has CDF for , for , for , and for . What is ?

Cumulative distribution functions Hard
A.
B.
C.
D.

46 For , a CDF is given by , and for . What is ?

Cumulative distribution functions Hard
A.
B.
C.
D.

47 A CDF satisfies and . It is continuous elsewhere near . What is the probability that equals ?

Cumulative distribution functions Hard
A.
B.
C.
D.

48 A continuous random variable has density for and otherwise. What are and ?

Probability density functions Hard
A.
B.
C.
D.

49 Let have density on . Define . What is the density of ?

Probability density functions Hard
A. for
B. for
C. for
D. for

50 Let for and otherwise. Which statement is correct?

Probability density functions Hard
A. Both the mean and median are infinite
B. The mean is infinite and the median is
C. The mean is and the median is
D. The mean is infinite and the median is

51 Random variables and satisfy , , and . What is ?

Mean and variance Hard
A.
B.
C.
D.

52 A random variable has and . What is ?

Mean and variance Hard
A.
B.
C.
D.

53 For , which expression equals ?

Binomial distribution Hard
A.
B.
C.
D.

54 If has mean and variance , what are and ?

Binomial distribution Hard
A.
B.
C.
D.

55 For , what is the probability that is even?

Binomial distribution Hard
A.
B.
C.
D.

56 Events occur according to a Poisson process. Given that exactly events occurred during an interval, what is the probability that none occurred during the first half of the interval?

Poisson distribution Hard
A.
B.
C.
D.

57 If , what is ?

Poisson distribution Hard
A.
B.
C.
D.

58 Let . Using , what is ?

Normal distribution Hard
A.
B.
C.
D.

59 Let and be independent. What is the distribution of ?

Normal distribution Hard
A.
B.
C.
D.

60 Four independent measurements are each distributed as . What is the probability that their maximum is at most ?

Normal distribution Hard
A.
B.
C.
D.
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