91Ó°ÊÓ

A box contains ten sealed envelopes numbered \(1, \ldots, 10\). The first five contain no money, the next three each contains \(\$ 5\), and there is a \(\$ 10\) bill in each of the last two. A sample of size 3 is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If \(X_{1}, X_{2}\), and \(X_{3}\) denote the amounts in the selected envelopes, the statistic of interest is \(M=\) the maximum of \(X_{1}, X_{2}\), and \(X_{3}\). a. Obtain the probability distribution of this statistic. b. Describe how you would carry out a simulation experiment to compare the distributions of \(M\) for various sample sizes. How would you guess the distribution would change as \(n\) increases?

Step by step solution

01

Understand the Distribution of Envelope Contents

First, let's understand how the envelopes are numbered and the amounts they contain: - Envelopes 1 to 5: $0 - Envelopes 6 to 8: $5 - Envelopes 9 to 10: $10 This means there are 5 envelopes with $0, 3 envelopes with $5, and 2 envelopes with $10.

02

Compute Probability of Each Amount for a Single Envelope

Calculate the probability of selecting each amount for a single trial:- Probability of \(0: \(P(X_i = 0) = \frac{5}{10} = 0.5\)- Probability of \)5: \(P(X_i = 5) = \frac{3}{10} = 0.3\)- Probability of $10: \(P(X_i = 10) = \frac{2}{10} = 0.2\)

03

Define the Maximum Statistic

For a sample size of 3 (with replacement), we are interested in the maximum value, \( M = \max(X_1, X_2, X_3) \). We will calculate the probability distribution of \( M \).

04

Calculate Probability \(M = 0\)

For \( M = 0 \), all three chosen envelopes must be $0. Thus, the probability is:\[ P(M = 0) = P(X_1 = 0, X_2 = 0, X_3 = 0) = (0.5)^3 = 0.125 \]

05

Calculate Probability \(M = 5\)

For \( M = 5 \), at least one but not all envelopes must contain \(5, and none can be \)10. The scenarios include having 1 or 2 envelopes with $5. Using the complement approach:\[ P(M = 5) = 1 - P(M = 0) - P(M = 10) \]First find \(P(M = 10)\) in the next step, then complete this calculation.

06

Calculate Probability \(M = 10\)

For \( M = 10 \), at least one envelope has to contain \(10. The complement of this is having all envelopes as \)0 or $5, which is \(1 - P(\text{all non-10})\):\[ P(\text{not 10 in 1 envelope}) = 0.8 \]Thus, \[ P(\text{all non-10}) = (0.8)^3 = 0.512 \]Finally,\[ P(M = 10) = 1 - 0.512 = 0.488 \]

07

Final Calculations for \(M = 5\)

As previously set up,\[ P(M = 5) = 1 - P(M = 0) - P(M = 10) = 1 - 0.125 - 0.488 = 0.387 \]

08

Simulation Experiment Description

To carry out a simulation, perform the following:1. Randomly select an envelope 3 times with replacement, recording the maximum value each time.2. Repeat this several times (e.g., 1000 times) to get a good approximation.3. Count the frequency of each maximum value to estimate the probability distribution.As \(n\) increases, the probability of getting higher amounts as maximum increases in the distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Sampling

Random sampling is a fundamental concept in probability and statistics. It refers to the process of selecting a subset of individuals from a larger set, where each individual has an equal chance of being chosen. In our scenario, you randomly select envelopes from a set of 10.
Since each envelope is sealed, and numbers are chosen with replacement, each selection is independent of previous ones. This means:

• All envelopes are equally likely to be selected during each draw.
• Each numerical value can be observed multiple times, because envelopes are replaced back into the box.
• Probabilities stay the same for each draw, maintaining a consistent `probability distribution` for each number within each sampling selection.

With random sampling, every subset of items from the set has the same probability of selection. In this experiment, this ensures fair assessment of the maximum statistic for each set of draws.

Simulation Experiment

Conducting a simulation experiment is a great way to approximate real-life outcomes. In this envelope exercise, a simulation helps in understanding the probability distribution of the maximum values (the maximum statistic) for different sample sizes.
A simulation requires several steps:

• First, randomly select envelopes multiple times (e.g., 1000 times) to mimic multiple real-world trials.
• Record the maximum value from each set of selected envelopes.
• Analyze the collected data to determine the frequency of each possible maximum value.

Simulating larger sample sizes helps illustrate changes in the distribution. As you increase the number of envelope selections, the likelihood of selecting a higher-max-value envelopes increases, demonstrating a shift in the maximum statistic as sample size grows.
This approach gives a clearer picture of how variable selections can affect outcomes, offering insights beyond theoretical calculations alone.

Envelope Probability

Envelope probability refers to the likelihood of selecting a certain amount of money from the envelopes. In this exercise scenario:
The event probabilities for selecting amounts when drawing an envelope are simple:

• The probability to select \(0 is \(0.5\) (or 50%), since there are 5 envelopes with \)0 out of 10.
• The probability to select \(5 is \(0.3\) (or 30%), since 3 envelopes contain \)5.
• The probability to select $10 is \(0.2\) (or 20%), with 2 envelopes containing this amount.

To find the probability of a maximum amount from three randomly chosen envelopes, you calculate probabilities for combinations, taking into account replacement.
These values are integral to understanding and predicting outcomes throughout both analytical probability assessments and simulation experiments.

Maximum Statistic

The maximum statistic in this exercise is defined as the largest value observed in a set of three envelope selections.
To calculate this, we assess all possible outcomes:

• If all three envelopes yield \(0, the maximum is obviously \)0; the probability is \(0.125\) (12.5%).
• If not all are \(0, at least one envelope should have a value of \)5, making some maximums \(5.
• For the maximum to be \)10, at least one envelope must be $10; this probability calculation shows \(0.488\) (or 48.8%).

The computations reveal how frequently each maximum amount is observed, assisting in understanding the empirical distribution results.
Importantly, the larger the sample size, the higher the chance of observing higher maximum values, illuminating how sample size impacts statistical behavior.