The Computer Selects A Number $X$ From 4 To 11 Randomly And Uniformly. Round All Answers To 4 Decimal Places Where Possible.a. What Is The Distribution Of $X$? - X ∼ U ( 4 , 11 X \sim U(4, 11 X ∼ U ( 4 , 11 ]b. Suppose That The Computer Randomly

Introduction

In probability theory, a uniform distribution is a type of continuous probability distribution where every possible outcome has an equal likelihood of occurring. In this article, we will explore the concept of a uniform distribution and apply it to a real-world scenario where a computer randomly selects a number between 4 and 11.

Understanding Uniform Distributions

A uniform distribution is characterized by a probability density function (PDF) that is constant over a given interval. In other words, the probability of each outcome is equally likely, and the distribution is flat over the interval. Mathematically, a uniform distribution can be represented as:

$$f(x) = \frac{1}{b-a} \quad \text{for } a \leq x \leq b$$

where $a$ and $b$ are the lower and upper bounds of the interval, respectively.

The Distribution of X

In our scenario, the computer selects a number $X$ from 4 to 11 randomly and uniformly. This means that the probability of selecting any number between 4 and 11 is equally likely. Therefore, the distribution of $X$ is a uniform distribution over the interval $[4, 11]$.

$$X \sim U(4, 11)$$

Properties of the Distribution

Since the distribution of $X$ is a uniform distribution, it has several properties that are worth noting:

• Constant Probability Density: The probability density function of $X$ is constant over the interval $[4, 11]$. This means that the probability of selecting any number between 4 and 11 is equally likely.
• Equal Probability of Outcomes: Since the distribution is uniform, the probability of each outcome is equally likely. In this case, the probability of selecting any number between 4 and 11 is $\frac{1}{11-4} = \frac{1}{7}$.
• Symmetry: The distribution of $X$ is symmetric around the midpoint of the interval $[4, 11]$, which is $\frac{4+11}{2} = 7.5$.

Calculating Probabilities

One of the key applications of uniform distributions is calculating probabilities. Since the distribution of $X$ is uniform, we can calculate the probability of selecting a number within a given interval.

For example, suppose we want to calculate the probability of selecting a number between 5 and 9. Since the distribution is uniform, we can use the following formula:

$$P(5 \leq X \leq 9) = \frac{9-5}{11-4} = \frac{4}{7}$$

Rounding Answers

As specified in the problem, we need to round all answers to 4 decimal places where possible. Therefore, the probability of selecting a number between 5 and 9 is approximately 0.5714.

Conclusion

In this article, we explored the concept of uniform distributions and applied it to a real-world scenario where a computer randomly selects a number between 4 and 11. We calculated the distribution of $X$ and its properties, including the constant probability density, equal probability of outcomes, and symmetry. We also demonstrated how to calculate probabilities using the uniform distribution. Finally, we rounded our answers to 4 decimal places where possible.

References

• [1] Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions. John Wiley & Sons.
• [2] Ross, S. M. (2010). Introduction to Probability Models. Academic Press.

Further Reading

For further reading on uniform distributions and probability theory, we recommend the following resources:

• [1] "Uniform Distribution" by Wolfram MathWorld
• [2] "Probability Theory" by Khan Academy
• [3] "Continuous Probability Distributions" by MIT OpenCourseWare
  The Computer Selects a Number: Q&A =====================================

Q: What is the probability of selecting a number between 4 and 11?

A: Since the distribution of $X$ is a uniform distribution over the interval $[4, 11]$, the probability of selecting any number between 4 and 11 is equally likely. Therefore, the probability of selecting a number between 4 and 11 is $\frac{1}{11-4} = \frac{1}{7} \approx 0.1429$.

Q: What is the probability of selecting a number greater than 7?

A: To calculate the probability of selecting a number greater than 7, we need to find the probability of selecting a number between 7 and 11. Since the distribution is uniform, we can use the following formula:

$$P(7 \leq X \leq 11) = \frac{11-7}{11-4} = \frac{4}{7} \approx 0.5714$$

Since we want to find the probability of selecting a number greater than 7, we need to subtract the probability of selecting a number less than or equal to 7 from 1. Therefore, the probability of selecting a number greater than 7 is:

$$P(X > 7) = 1 - P(X \leq 7) = 1 - \frac{7-4}{11-4} = 1 - \frac{3}{7} = \frac{4}{7} \approx 0.5714$$

Q: What is the probability of selecting a number less than 5?

A: To calculate the probability of selecting a number less than 5, we need to find the probability of selecting a number between 4 and 5. Since the distribution is uniform, we can use the following formula:

$$P(4 \leq X \leq 5) = \frac{5-4}{11-4} = \frac{1}{7} \approx 0.1429$$

Since we want to find the probability of selecting a number less than 5, we need to subtract the probability of selecting a number greater than or equal to 5 from 1. Therefore, the probability of selecting a number less than 5 is:

$$P(X < 5) = 1 - P(X \geq 5) = 1 - \frac{11-5}{11-4} = 1 - \frac{6}{7} = \frac{1}{7} \approx 0.1429$$

Q: What is the expected value of $X$?

A: The expected value of $X$ is the average value of the distribution. Since the distribution is uniform, the expected value of $X$ is the midpoint of the interval $[4, 11]$, which is:

$$E(X) = \frac{4+11}{2} = 7.5$$

Q: What is the variance of $X$?

A: The variance of $X$ is a measure of the spread of the distribution. Since the distribution is uniform, the variance of $X$ is:

$$Var(X) = \frac{(11-4)^2}{12} = \frac{49}{12} \approx 4.0833$$

Q: What is the standard deviation of $X$?

A: The standard deviation of $X$ is the square root of the variance. Therefore, the standard deviation of $X$ is:

$$SD(X) = \sqrt{Var(X)} = \sqrt{\frac{49}{12}} \approx 1.5895$$

Conclusion

In this article, we answered several questions related to the uniform distribution of $X$. We calculated the probability of selecting a number between 4 and 11, the probability of selecting a number greater than 7, the probability of selecting a number less than 5, the expected value of $X$, the variance of $X$, and the standard deviation of $X$. We also provided explanations and formulas for each calculation.