Q: Suppose you want to minimize the expected loss in predicting whether a flight is late, where the loss function is $$L = (1 - p)^2$ if the flight is late and p^2$ if it is not late. What value of p$ minimizes the expected loss?

$$p = P(\text{flight is late})$$

Question 1

Which of the following best describes the Box-Muller method for generating standard normal random variables?

Accepted Answer

It transforms two independent uniform random variables into two independent standard normal random variables using polar coordinates.

Question 2

Suppose $$U_1$$ and $$U_2$$ are independent random variables uniformly distributed on $$(0,1). $$According to the Box-Muller method, which of the following formulas correctly generates a standard normal random variable $$X$$?

Accepted Answer

$$X = \sqrt{-2 \ln U_1$} \cos(2\pi U_2$)$$

Question 3

In a Monte Carlo study, you want to estimate the mean $$\mu of a $$distribution. If the variance of your estimator is $$\$$sigma^2$$/n$$, how many simulation runs $$n$$ are needed to ensure that your estimate is within $$\pm \epsilon of$$ $$\mu$$ with 99% confidence?

Accepted Answer

$$n = \dfrac{2.58^2$ \$$sigma^2$$}{\$$epsilon^2$$}$$

Question 4

Which of the following is a key assumption of the naive Bayes approach in machine learning?

Accepted Answer

All features are independent given the class label.

Question 5

Given a historical database of flight delays, the naive Bayes estimate for the probability that a flight with vector $$x = (x_1$, x_2$, ..., x_n) is $$late is:

Accepted Answer

$$P(L|x) = \dfrac{P(x|L)P(L)}{P(x)}$$

Question 6

Suppose you have a dataset with $$n$$ independent and identically distributed random variables $$X_1$$, $$X_2$$, ..., X_n$$. The bootstrap estimate of $$Var(S^2$$)$$, where $$S^2 is $$the sample variance, is given by:

Accepted Answer

$$p = \dfrac{1}{B} \sum_{b=1}^B (S_b^2$ - \bar{S}^2)^2$

Question 7

In the $$k-$$nearest neighbors rule, how is the distance between two characterizing vectors $$x$$ and $$y$$ typically defined?

Accepted Answer

$$d(x, y) = \sum_{i=1}^n |x_i - y_i|$$

Question 8

Which of the following statements about logistic regression is correct?

Accepted Answer

It estimates the probability of a binary outcome as $$P(Y=1|x) = \dfrac{e$$^{\beta_0 + \beta_1 x_1 + ... + \beta_n x_n}$$}{1 + e$$^{\beta_0 + \beta_1 x_1 + ... + \beta_n x_n}$$}$$

Question 9

Suppose you want to minimize the expected loss in predicting whether a flight is late, where the loss function is $$L = (1 - p)^2$ if the flight is late and p^2$ if it is not late. What value of p$ minimizes the expected loss?

Accepted Answer

$$p = P(	ext{flight is late})$$

Question 10

In the context of the bandit problem, if drug 1 has been used 55 times with 35 successes and drug 2 has been used 25 times with 12 successes, what is the estimated probability of success for drug 1?

Accepted Answer

$$\dfrac{35}{55}$$

Question 11

Which of the following is a real-world application of the naive Bayes classifier?

Accepted Answer

Predicting whether an email is spam based on the presence of certain words.

Question 12

Which of the following best describes the purpose of using bootstrap methods in statistics?

Accepted Answer

To estimate the sampling distribution of a statistic by resampling with replacement from the observed data.

Question 13

Suppose you have a set of $$n$$ independent and identically distributed random variables $$X_1$$, ..., X_n$$ with mean $$\mu$$ and variance $$\$$sigma^2. $$The sample mean $$\bar{X} is an $$unbiased estimator of $$\mu. $$What is the variance of $$\bar{X}$$?

Accepted Answer

$$\dfrac{\$$sigma^2$$}{n}$$

Question 14

In the context of machine learning, what is the main advantage of using the $$k-$$nearest neighbors rule?

Accepted Answer

It makes no assumptions about the underlying distribution of the data.

Question 15

Which of the following is true about the loss function $$L = (1-p)^2$ if the event occurs and L$$ = $$p^2 if it $$does not occur?

Accepted Answer

The expected loss is minimized when $$p$$ equals the true probability of the event.

Question 16

Which of the following is a limitation of the naive Bayes classifier?

Accepted Answer

It assumes conditional independence of features given the class label, which may not hold in practice.

Question 17

Suppose you want to estimate the probability that a flight is late using a distance-weighted method. Which historical flights should be given more weight?

Accepted Answer

Flights whose characterizing vectors are closer to the flight under consideration.

Question 18

Which of the following is the correct formula for the logistic regression probability model?

Accepted Answer

$$P(Y=1|x) = \dfrac{e$$^{\beta_0 + \beta_1 x_1 + ... + \beta_n x_n}$$}{1 + e$$^{\beta_0 + \beta_1 x_1 + ... + \beta_n x_n}$$}$$

Simulation, Bootstrap Methods, Permutation Tests, and Machine Learning in Statistics

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