Unit 5 - Practice Quiz

1 What is the expected value of a discrete random variable ?

2 If the expected value of a random variable is , what is ?

3 What is the expected value of a constant ?

4 Which of the following formulas correctly represents the linearity of expectations for any two random variables and ?

5 If and , what is the value of ?

6 Does the property require the random variables and to be independent?

7 Which of the following is the defining formula for the variance of a random variable ?

8 What is the alternative (and often easier to compute) formula for calculating the variance ?

9 Can the variance of a real-valued random variable be negative?

10 Two discrete random variables and are independent if and only if for all possible values and :

11 If and are independent random variables, what does equal?

12 If two random variables and are independent, what is their covariance?

13 What is the standard formula for Bayes' Theorem used to find ?

14 In Bayes' theorem, what is typically called before observing evidence ?

15 The denominator in the generalized Bayes' theorem, , represents what probability by the Law of Total Probability?

16 In the context of Bayes' theorem, what does represent?

17 What does a geometric random variable typically model?

18 If the probability of success in a single trial is , what is the expected value of a geometric random variable?

19 If a biased coin has a probability of $0.25$ of landing on heads, what is the expected number of flips needed to get the first head?

20 According to Bienaymé's formula, the variance of the sum of pairwise independent random variables is equal to:

21 Two machines, A and B, manufacture 40% and 60% of the total items produced in a factory, respectively. Machine A has a defect rate of 3%, and Machine B has a defect rate of 5%. If a randomly selected item is found to be defective, what is the probability it was produced by Machine A?

22 A diagnostic test has a 90% true positive rate and a 10% false positive rate. If the disease prevalence in the population is 5%, what is the probability that a person actually has the disease given that they test positive?

23 Suppose there are three identical boxes. Box 1 contains only gold coins, Box 2 contains only silver coins, and Box 3 contains half gold and half silver coins. You randomly select a box and draw a coin. If the coin is gold, what is the probability you selected Box 1?

24 Consider a game where you roll a fair 6-sided die. If the die lands on an even number, you win that number of dollars. If the die lands on an odd number, you lose that number of dollars. What is the expected value of your winnings per roll?

25 A random variable takes the values $0, 1$, and $2$ with probabilities $0.2, 0.5$, and $0.3$ respectively. What is the expected value of ?

26 A coin is weighted such that it is twice as likely to land on Heads as it is to land on Tails. If it lands on Heads, you gain 3 points. If it lands on Tails, you lose 3 points. What is the expected point gain from one flip?

27 What is the expected sum of the numbers appearing on three fair 6-sided dice rolled simultaneously?

28 A biased coin with a probability of landing heads is flipped 5 times. Let be the number of heads and be the number of tails. What is ?

29 You draw 5 cards from a well-shuffled standard deck of 52 cards without replacement. What is the expected number of Aces in your hand?

30 You roll a fair 6-sided die repeatedly. What is the expected number of rolls until you observe a 6?

31 The probability of a successful login attempt to a server is 0.8. Assuming independent attempts, what is the probability that exactly 3 attempts are needed to login successfully?

32 Let representing the number of trials up to and including the first success. If , what is the probability that ?

33 Let and be independent random variables with and . If , what is the value of ?

34 Let and be the outcomes of two independent fair 6-sided die rolls. What is the probability that ?

35 Let and be independent random variables taking values in . If and , what is the probability that ?

36 Let be the outcome of a single roll of a fair 6-sided die. What is the variance of ?

37 If a random variable has a variance of $5$, what is the variance of the random variable ?

38 A discrete random variable has an expected value and the expected value of its square is . What is the standard deviation of ?

39 Let be pairwise independent random variables, each with a variance of $4$. What is the variance of their sum, ?

40 Let and be pairwise independent random variables with and . What is the variance of the random variable ?

41 A rare disease affects of a population. A diagnostic test for this disease has a true positive rate and a false positive rate. If a person takes the test twice and the results are independent given the disease status, what is the probability they actually have the disease given that both tests return positive?

42 A monkey types uniformly at random on a standard 26-letter English alphabet keyboard. What is the expected number of keystrokes until the sequence 'MATH' appears for the first time?

43 Consider a random permutation of the set where . A position () is a local maximum if and . Positions $1$ and are local maxima if and , respectively. What is the expected number of local maxima in the permutation?

44 Alice and Bob take turns flipping a biased coin that lands on heads with probability . Alice flips first. The first person to flip heads wins. What is the probability that Alice wins the game?

45 Let and be independent Poisson random variables with parameters and , respectively. What is the conditional distribution of given that ?

46 Let be the number of fixed points in a uniformly random permutation of elements (). What is the variance of ?

47 Bienaymé's formula states that the variance of the sum of random variables is equal to the sum of their variances. What is the strictly minimal requirement on the random variables for this formula to hold?

48 A spam filter is designed such that of emails are spam. It is known that of spam emails contain the word 'lottery', while only of non-spam (ham) emails contain this word. If an incoming email contains the word 'lottery', what is the probability that it is spam?

49 Consider a random graph where each edge is included independently with probability . What is the expected number of isolated vertices in the graph?

50 If balls are placed uniformly and independently at random into bins, what is the expected number of empty bins?

51 Let and be independent geometric random variables, both representing the number of trials up to and including the first success, with probability of success . What is ?

52 Let and be independent uniform random variables over the set . What is the probability distribution of ?

53 Let be a random variable with mean and variance . What value of strictly minimizes the expected value , and what is this minimum expected value?

54 Let be independent random variables where each represents the outcome of a fair -sided die roll (values $1$ through ). Using Bienaymé's formula, what is the variance of their sum ?

55 If there are people in a room, what is the expected number of pairs of people who share the exact same birthday? (Assume a 365-day year and independent uniform birthday probabilities)

56 A student guesses completely randomly on a multiple-choice exam with questions. Each question has options, and exactly one is correct. If the student gets point for a correct answer and points for an incorrect answer, what is their expected total score?

57 Let be a geometric random variable with success probability , representing the number of trials up to and including the first success. Due to the memoryless property, what is the conditional expectation ?

58 Let and be independent random variables. Which of the following statements is NOT necessarily true?

59 Three identical-looking urns contain colored balls. Urn 1 has 2 Red and 1 Black; Urn 2 has 1 Red and 2 Black; Urn 3 has 3 Red and 0 Black. An urn is selected uniformly at random, and TWO balls are drawn WITH replacement from it. Both balls are Red. What is the probability that Urn 1 was chosen?

60 Let be the number of trials until the -th success occurs in a sequence of independent Bernoulli trials with probability of success (a Negative Binomial distribution). What is the variance of ?