Probability

1.A condition is present in 1 percent of names. A screen catches 99 percent of those and also flags 5 percent of the names without the condition. A name is flagged. What fraction of flagged names have the condition?

2.Two research notes fail independently, each with probability 0.2. What is the probability that at least one fails?

3.A coin is fair. You toss until the first head. What is the expected number of tosses?

4.A die is fair and six-sided. What is the expected face value?

5.X and Y are dependent, and each has a finite mean. Which sum is valid for the pair?

6.Two independent books each have variance 4. What is the variance of their sum?

7.A return is multiplied by 3, as when a book is levered three times. What happens to its variance?

8.The sample average of independent draws from a distribution with mean mu uses the usual 1/n weights. What is true of that average as an estimator of mu?

9.A researcher divides the sum of squared deviations from the sample mean by n, not by n−1, and uses that as the variance. For an independent sample from a finite variance, what is that divisor doing?

10.A test's p-value is 0.03 under a preset null. Which reading matches the definition?

11.A method produces a 95 percent interval for a mean from each new sample. Across many samples, what does 95 percent refer to?

12.X is uniform on {−1, +1} and Y = X squared. The covariance of X and Y is zero. Are X and Y independent?

13.Three cards: both faces red, both faces black, and one of each. A card is drawn at random and a random face is red. What is the probability the other face is red?

14.Three sealed funds, one with a real record. You pick one. The host, who knows the records, opens a different fund that is fake. If you switch to the remaining fund, what is your probability of the real record?

15.You need both a head and a tail from a fair coin, and you toss until you have seen both. What is the expected number of tosses?

16.Draws are independent and uniform on [0, 1]. N is the first count whose draws sum to more than 1. What is E[N]?

17.A fair coin is tossed and you stop at the first time you are ahead by one dollar. You stake one dollar each toss. What is the expected number of tosses?

18.A 95 percent interval is built from a sample of n independent draws. If n is multiplied by 4 and nothing else changes, how does the interval's width scale?

19.A book of independent bets, each with win probability p, is sized so that a win pays even money. The growth-optimal fraction of wealth bet on one such bet is the Kelly fraction. For p = 0.6, what fraction is that?

20.Events A and B each have probability 0.5, and P(A and B) = 0.2. What is P(A or B)?

21.In a random permutation of n distinct research memos, the probability that the memo in position n is a record high, meaning the largest among the first n, is what?

22.The number of fixed points in a random permutation of a large number of names is approximately Poisson with which mean?

23.Two people arrive at uniform random times in an hour and each waits 15 minutes. What is the probability they meet?

24.A fair game pays +1 or −1. You must stop at a fixed toss, chosen before any toss. What is the expected fortune at that toss?

25.Scores are skewed right. A program quotes the mean as the typical score. Which comparison is the usual one for a right skew?

26.An indicator of an event with probability p has what variance?

27.Independent Poisson counts of rates 2 and 5 are added. The sum is Poisson with what rate?

28.A Poisson process of rate 4 is thinned by keeping each point independently with probability 1/4. What is the rate of the kept points?

29.Given that a Poisson count of rate 6 equals 1, the single event's time in the interval is uniform. More useful here: the conditional law of one independent Poisson given the sum. Two independent Poissons of rates 1 and 3 sum to 4. Given the sum, the first count is which distribution?

30.A tower of expectations says E[E[X given Y]] equals what?

31.You and a rival toss a fair coin. You start with 3 dollars, the rival with 7, and each toss moves 1 dollar. Play stops at 0 or 10. What is your probability of reaching 10?

32.A concave utility represents a risk-averse desk. For a random payoff X with a finite mean, Jensen's inequality says which comparison?

33.Last season's top fund, among many, had a huge result that was mostly noise around the same true mean. What does regression to the mean predict for next season?

34.A treatment looks helpful in each of two sectors and harmful once the sectors are pooled. What is that pattern called, and what drives it?

35.Buses arrive every 10 minutes on a fixed schedule. A rider appears at a random moment. What waiting time do they average?

36.Bus gaps are random, some short and some long, with mean gap 10 minutes. A rider arrives at a random time and is more likely to land in a long gap. How does their expected gap compare with 10 minutes?

37.The standard deviation of an average of n independent draws, each of standard deviation s, is what?

38.A constant is added to every return. What happens to the variance?

39.Every return is multiplied by −1. What happens to the variance?

40.The covariance of a random return with a constant cash yield is what?

41.For a standard normal, the mean of the positive part, E[Z given Z > 0], equals what?

42.An exponential waiting time with mean 10 has already lasted 10. What is the expected remaining wait?

43.Two independent fair bits are 1 with probability 1/2. Let X be the first, Y the second, and Z = X xor Y, the sum mod 2. Which independence statement is true?

44.A forecast f of X is measurable with respect to the information you have. Among such forecasts, the conditional mean minimizes which quantity?

45.A desk reports that 70 percent of the names it called 70 percent likely did occur. What property is that?

46.Odds of 3 to 1 in favor of an event are which probability?

47.A rare condition has probability near zero, and a screen's false-positive rate is much larger than the base rate. The positive predictive value is then closest to which simplification?

48.Sensitivity is the probability of a flag given the condition. It is not which of the following?

49.Chebyshev's inequality says that a variable with mean mu and standard deviation s falls more than k standard deviations from mu with probability at most what?

50.The central limit theorem, in the usual form, is a statement about what?

51.A sample variance from normal data, scaled the usual way, is chi-squared. For a badly skewed population, a sample of size 10, and a claim that the sample mean is already normal enough to quote a tight normal interval, what is the cautious reading?

52.The law of large numbers says that the sample average of independent identically distributed draws with a finite mean does what as n grows?

53.A maximum-likelihood estimator is invariant under a one-to-one relabeling of the parameter. If theta-hat maximizes the likelihood, the maximum-likelihood estimate of g(theta) for a one-to-one g is what?

54.A Bayesian update starts from a prior mean m and a prior variance, and the likelihood is a normal sample. The posterior mean is a weighted average of m and the sample mean. If the sample is small, that posterior mean lies where?

55.St. Petersburg: a fair coin is tossed until the first head, on toss k, and the payoff is 2^k dollars. What is the expected payoff?

56.The same St. Petersburg payoff is valued with a log utility. The expected utility is finite. What does that show?

57.The expected number of uniform{1,...,n} coupons needed to collect all n types, for large n, grows like what?

58.P(A | B) is 0.9 and P(B) is 0.2. What is P(A and B)?

59.P(A | B) is 0.8 and P(B | A) is 0.4. Can you recover P(A) from these two numbers alone?

60.Five independent screens each have a 0.1 chance of a false alarm, and there is no true case. What is the probability of at least one false alarm?

61.A correlation is 1.2 in a printed table. What should you conclude?

62.R-squared of a regression is 0.64. Which reading is fair?

63.Two unbiased estimators of the same mean are independent, with variances 1 and 4. The minimum-variance weighted average puts what weight on the more precise one?

64.The same two estimators are averaged with equal weights instead. How does that average's variance compare with the inverse-variance blend?

65.A single draw from a normal mean is unbiased for the mean. Averaging n independent draws cuts the variance by n. Which statement about one draw is right?

66.You square a mean-zero random variable to estimate its variance from one draw, using X squared. For a normal, that estimator is unbiased for the variance. It is also what, relative to the usual sample variance from many draws?

67.Pairs (X, Y) are sampled only when X + Y is positive. Both coordinates are standard normal and independent in the population. In the selected sample, the correlation of X and Y is what?

68.A trader looks at the best of 20 independent noise series, each mean zero, and treats the winner's in-sample mean as an estimate of its true mean. What is the expectation of the winner's sample mean?

69.The probability that two independent uniform{1,...,6} dice show the same face is what?

70.The probability that the sum of two fair dice is 7 is what?

71.A fair die is rolled twice. What is the expected value of the larger face?

72.P(the maximum of two fair dice is at most 2) is what?

73.An event has been observed n times in a row under independent trials with unknown probability p. The maximum-likelihood estimate of p is 1. A Bayesian with a uniform prior on p has posterior mean what, after n successes and no failures?

74.The same uniform prior sees s successes and f failures. The posterior mean of p is what?

75.Two analysts give probabilities 0.2 and 0.7 for the same event, and only one can be the desk's quoted price. The event happens. A log score charges minus the log of the quoted probability. Who scores better?

76.A quoted probability of 0 for an event that then occurs gives a log score of what?

77.X and Y have correlation 0.5, and each has standard deviation 2. What is their covariance?

78.The covariance is 2 and both standard deviations are 2. If both variables are multiplied by 10, what is the new correlation?

79.A regression residual, in a model with an intercept fitted by least squares, has sample correlation with the fitted values equal to what?

80.The variance of a sum of two variables equals the sum of the variances. What else is true?

81.You estimate a mean with an average. You then square the standard error and call it the variance of a single future draw. What did you compute instead?

82.A 99 percent interval and a 90 percent interval are built from the same sample and the same model. Which is wider?

83.Independent trials with success probability 0.25. The expected number of successes in 40 trials is what?

84.The variance of that binomial count, n = 40 and p = 0.25, is what?

85.A geometric number of failures before the first success, with success probability p, has mean (1 − p)/p. For p = 0.2, the expected number of failures before the first success is what?

86.Memorylessness of the geometric distribution says that, given you have already seen 6 failures, the additional failures you still expect equal what, for the same p?

87.Two events are mutually exclusive and each has probability 0.4. What is the probability that neither occurs?

88.If two events are mutually exclusive and both have positive probability, they are not independent. Why?

89.A likelihood ratio of 1, new data against a point null, means what about the data's evidence?

90.You rank n independent continuous noisy scores and keep the top one as a signal. As n grows and the true means stay equal, the expected score of the one you keep does what?

91.A probability forecast is coherent when a book of bets on it cannot lock in a sure loss. A quoted set of probabilities that sums to 1.4 on a partition of three outcomes fails coherence because of what?

92.F is the cumulative distribution function of a continuous return, and U = F(X). If F is the true distribution, U is distributed as what?

93.A forecast distribution is too narrow, and you transform realized returns by that forecast's distribution function. Those transformed values, compared with a uniform, will tend to do what?

94.The median of a sample of continuous data minimizes the expected absolute deviation from a point forecast. The mean minimizes the expected squared deviation. A desk that is charged the absolute error, and quotes the mean of a skewed payoff, is optimizing what?

95.A stop-loss on a fair symmetric random walk is placed at a fixed loss, and a take-profit at the same distance on the other side. You stop at whichever is hit first. What is the expected stopped value?

96.The same fair walk has a take-profit at +1 and no stop on the downside, and you claim the expected value at stopping is +1 because you almost surely get there. What else is true of the waiting time?

97.n independent hypothesis tests each have type I error 0.05, and all nulls are true. The expected number of false rejections is what?

98.Those n tests are independent and all nulls are true. The probability of no false rejection is what?

99.A covariance matrix of three returns has a negative entry on the diagonal in a spreadsheet. What is wrong?

100.A correlation matrix must be positive semidefinite, and its diagonal is all ones. A 2×2 matrix with off-diagonal 1.1 fails because of what?

101.You simulate a strategy's terminal wealth by compounding independent daily returns that have a positive mean and a large volatility. The mean of terminal wealth and the median of terminal wealth typically compare how, over a long horizon?

102.The log of wealth is an average of log gross returns. If the expected simple return is positive and the expected log return is negative, a typical long path does what?

103.A researcher replaces each probability in a discrete distribution by its square and renormalizes. Starting from a two-point distribution with probabilities 0.8 and 0.2, the new weight on the first point is what?

104.A Cauchy law, the ratio of two independent standard normals, has no mean. Averaging independent copies of it does what to the average's distribution?

105.A researcher wins a point with probability 0.6, independently, and a set is first to 2 points. What is the probability the researcher wins the set?

106.A Weibull-type wait with an increasing hazard, compared with an exponential of the same mean, has already lasted a long time. The expected remaining wait is then what, relative to a fresh exponential wait?

107.K independent forecasts of the same event have errors with mean zero and equal variance, and the errors are uncorrelated. The variance of the average of the K forecasts, as an estimator of the target, is the single-forecast error variance divided by what?

108.Those forecast errors are perfectly correlated instead. The variance of the average is then what, relative to one forecast?