Problem 2 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

We generate a bootstrap dataset \(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}, x_{4}^{*}\) from the empirical distribution function of the dataset \(\begin{array}{llll}1 & 3 & 4 & 6 .\end{array}\) a. Compute the probability that the bootstrap sample mean is equal to \(1 .\) b. Compute the probability that the maximum of the bootstrap dataset is equal to 6 . c. Compute the probability that exactly two elements in the bootstrap sample are less than \(2 .\)

Step-by-Step Solution

Answer

a. \( \frac{1}{256} \); b. \( \frac{175}{256} \); c. \( \frac{27}{128} \).

Step 1: Understand the Empirical Distribution Function

The empirical distribution function of a dataset is a probability distribution that assigns equal probability to each element in the dataset. For our dataset \( \{1, 3, 4, 6\} \), each element has a probability of \( \frac{1}{4} \) of being selected in the bootstrap sample.

Step 2: Calculate Probability of Bootstrap Sample Mean Equal to 1

Since the mean of a sample cannot be less than the minimum value in the sample, for the mean to be 1, all selected bootstrap values must be 1. The probability of selecting 1 for each sample point is \( \frac{1}{4} \). Therefore, the probability that \( x_1^* = x_2^* = x_3^* = x_4^* = 1 \) is \( \left( \frac{1}{4} \right)^4 = \frac{1}{256} \).

Step 3: Calculate Probability of Bootstrap Maximum Equal to 6

For the maximum to be 6, at least one of the sampled elements must be 6. The probability of not choosing 6 in any one trial is \( \frac{3}{4} \), so for four trials, it is \( \left( \frac{3}{4} \right)^4 \). Therefore, the probability of at least one element being 6 (i.e., the maximum being 6) is \( 1 - \left( \frac{3}{4} \right)^4 = \frac{175}{256} \).

Step 4: Calculate Probability of Exactly Two Elements Less than 2

Elements less than 2 in the dataset are \{1\}. Find the probability of choosing 1 exactly twice from 4 positions. The probability of choosing 1 in a position is \( \frac{1}{4} \), and choosing not 1 (3, 4, or 6) is \( \frac{3}{4} \). The required probability is given by binomial probability: \( \binom{4}{2} \left( \frac{1}{4} \right)^2 \left( \frac{3}{4} \right)^2 = 6 \times \frac{1}{16} \times \frac{9}{16} = \frac{27}{128} \).

Key Concepts

Empirical Distribution Function

The empirical distribution function (EDF) is a crucial concept in statistics for understanding the nature of your data. Imagine you have a dataset with elements, as in our case: \( \{1, 3, 4, 6\} \). The EDF is simply a step function that increases by \( \frac{1}{n} \) at each of the \( n \) data points.

• Each data point in the distribution, therefore, has an equal probability of being chosen. In our dataset, each number has a \( \frac{1}{4} \) chance of selection.
• This function helps in generating bootstrap samples, which replicate the sampling distribution by repeatedly drawing samples from the data.

By using the EDF, we make the assumption that the observed sample is a good approximation of the actual population from which the sample was generated. It effectively treats the observed sample as a representation of the entire dataset.

Bootstrap Sample Mean

The bootstrap sample mean is a way to estimate the average of a bootstrapped dataset. In bootstrap sampling, we repeatedly resample data with replacement.

• For instance, to find the probability that the bootstrap sample mean equals 1, all elements in the sample must be 1.
• Because each element in the dataset has a \( \frac{1}{4} \) chance of being picked, the probability that every selected value is 1 is \( \left( \frac{1}{4} \right)^4 \) or \( \frac{1}{256} \).

Calculating the bootstrap sample mean helps us understand the variability of our data. This technique is useful in real-world scenarios where we want to estimate the distribution of the sample means without making heavy assumptions about the population distribution.

Bootstrap Maximum

The term bootstrap maximum refers to the largest value in a bootstrapped dataset derived from an empirical distribution. To determine the probability that the maximum of the sample is 6, you must consider that at least one of the draws is a 6.

• The probability that a particular draw does not result in a 6 is \( \frac{3}{4} \), because there are three other numbers (1, 3, 4).
• The probability that none of the four draws results in a 6 is \( \left( \frac{3}{4} \right)^4 \).
• Thus, to find the probability that at least one is 6, we calculate \( 1 - \left( \frac{3}{4} \right)^4 \), which equals \( \frac{175}{256} \).

In bootstrap analysis, understanding the maximum value is important because it can impact the interpretation of your data's range and variability.

Binomial Probability

Binomial probability comes into play when you want to determine the chances of a particular number of successes in a sequence of independent experiments. In our example, we look at the probability of having exactly two elements in a bootstrap sample that are less than 2.

• The only element less than 2 in the dataset is 1, and its probability is \( \frac{1}{4} \).
• To compute the probability of exactly two ones being selected, we use the binomial coefficient: \( \binom{4}{2} \), which counts how many ways we can choose 2 successes out of 4 trials.
• The full probability calculation is \( \binom{4}{2} \left( \frac{1}{4} \right)^2 \left( \frac{3}{4} \right)^2 \), giving us \( \frac{27}{128} \).

This calculation is crucial in statistics because it provides the likelihood of observing a specific combination of outcomes, which supports decision-making and predictions based on data.