12. A Continuous Random Variable \[$X\$\] Has A Cumulative Distribution Function Given By:$\[ F(x) = \begin{cases} \frac{1+x}{6}, & -1 \ \textless \ X \ \textless \ 0 \\ \frac{1+2x}{6}, & 0 \ \textless \ X \ \textless \ 2

Introduction

In probability theory, a cumulative distribution function (CDF) is a fundamental concept used to describe the probability distribution of a random variable. It provides a way to calculate the probability that a random variable takes on a value less than or equal to a given value. In this article, we will explore the cumulative distribution function of a continuous random variable, denoted as $X$, and examine its properties.

The Cumulative Distribution Function

The cumulative distribution function of the continuous random variable $X$ is given by:

${ F(x) = \begin{cases} \frac{1+x}{6}, & -1 \ \textless \ x \ \textless \ 0 \\ \frac{1+2x}{6}, & 0 \ \textless \ x \ \textless \ 2 \end{cases} }$

This function is defined in two parts: one for the interval $-1 < x < 0$ and another for the interval $0 < x < 2$. We will examine each part separately.

Properties of the Cumulative Distribution Function

To understand the properties of the cumulative distribution function, we need to examine its behavior in different intervals. Let's start with the interval $-1 < x < 0$.

Interval $-1 < x < 0$

In this interval, the cumulative distribution function is given by:

${ F(x) = \frac{1+x}{6} }$

To examine the behavior of this function, let's calculate its value at the endpoints of the interval.

• At $x = -1$, we have:

${ F(-1) = \frac{1+(-1)}{6} = \frac{0}{6} = 0 }$

• At $x = 0$, we have:

${ F(0) = \frac{1+0}{6} = \frac{1}{6} }$

As we can see, the cumulative distribution function is equal to 0 at the left endpoint of the interval and equal to $\frac{1}{6}$ at the right endpoint.

Interval $0 < x < 2$

In this interval, the cumulative distribution function is given by:

${ F(x) = \frac{1+2x}{6} }$

To examine the behavior of this function, let's calculate its value at the endpoints of the interval.

• At $x = 0$, we have:

${ F(0) = \frac{1+2(0)}{6} = \frac{1}{6} }$

• At $x = 2$, we have:

${ F(2) = \frac{1+2(2)}{6} = \frac{5}{6} }$

As we can see, the cumulative distribution function is equal to $\frac{1}{6}$ at the left endpoint of the interval and equal to $\frac{5}{6}$ at the right endpoint.

Conclusion

In this article, we have examined the cumulative distribution function of a continuous random variable, denoted as $X$. We have seen that the function is defined in two parts: one for the interval $-1 < x < 0$ and another for the interval $0 < x < 2$. We have also examined the properties of the function in each interval and calculated its value at the endpoints of the intervals.

Key Takeaways

• The cumulative distribution function of a continuous random variable is a fundamental concept used to describe the probability distribution of the variable.
• The function is defined in two parts: one for the interval $-1 < x < 0$ and another for the interval $0 < x < 2$.
• The function has different properties in each interval, and its value at the endpoints of the intervals is equal to 0, $\frac{1}{6}$, and $\frac{5}{6}$.

Further Reading

For further reading on cumulative distribution functions, we recommend the following resources:

• "Probability Theory" by E.T. Jaynes
• "Statistics for Dummies" by Deborah J. Rumsey
• "Probability and Statistics" by James E. Gentle

References

• Jaynes, E.T. (2003). Probability Theory: The Logic of Science. Cambridge University Press.
• Rumsey, D.J. (2011). Statistics for Dummies. John Wiley & Sons.
• Gentle, J.E. (2006). Probability and Statistics. Springer.
  Frequently Asked Questions: Cumulative Distribution Functions =================================================================

Q: What is a cumulative distribution function?

A: A cumulative distribution function (CDF) is a function that describes the probability distribution of a random variable. It provides a way to calculate the probability that a random variable takes on a value less than or equal to a given value.

Q: What is the purpose of a cumulative distribution function?

A: The purpose of a cumulative distribution function is to provide a way to calculate the probability of a random variable taking on a value within a given range. This is useful in a variety of applications, including statistics, engineering, and finance.

Q: How is a cumulative distribution function defined?

A: A cumulative distribution function is defined as the probability that a random variable takes on a value less than or equal to a given value. It is typically denoted as F(x) and is defined as:

${ F(x) = P(X \leq x) }$

Q: What are the properties of a cumulative distribution function?

A: A cumulative distribution function has several properties, including:

• Non-decreasing: The CDF is non-decreasing, meaning that as the value of x increases, the value of F(x) also increases.
• Right-continuous: The CDF is right-continuous, meaning that as the value of x approaches a certain value from the right, the value of F(x) approaches the same value.
• Bounded: The CDF is bounded between 0 and 1, meaning that 0 ≤ F(x) ≤ 1 for all values of x.

Q: How do I calculate the cumulative distribution function of a random variable?

A: To calculate the cumulative distribution function of a random variable, you need to know the probability distribution of the variable. The CDF can be calculated using the following formula:

${ F(x) = \int_{-\infty}^{x} f(t) dt }$

where f(t) is the probability density function (PDF) of the random variable.

Q: What is the difference between a cumulative distribution function and a probability density function?

A: A cumulative distribution function (CDF) and a probability density function (PDF) are related but distinct concepts. The CDF describes the probability distribution of a random variable, while the PDF describes the rate at which the probability distribution changes.

Q: How do I use a cumulative distribution function in real-world applications?

A: Cumulative distribution functions are used in a variety of real-world applications, including:

• Statistics: CDFs are used to calculate probabilities and percentiles in statistical analysis.
• Engineering: CDFs are used to design and optimize systems, such as bridges and buildings.
• Finance: CDFs are used to calculate risk and return in financial modeling.

Q: What are some common types of cumulative distribution functions?

A: Some common types of cumulative distribution functions include:

• Uniform distribution: The CDF of a uniform distribution is a straight line.
• Normal distribution: The CDF of a normal distribution is a cumulative normal distribution function.
• Exponential distribution: The CDF of an exponential distribution is a cumulative exponential distribution function.

Q: How do I choose the right cumulative distribution function for my application?

A: To choose the right cumulative distribution function for your application, you need to consider the following factors:

• Data distribution: The CDF should match the distribution of your data.
• Application requirements: The CDF should meet the requirements of your application, such as calculating probabilities and percentiles.

Conclusion

In this article, we have answered some frequently asked questions about cumulative distribution functions. We have discussed the definition, properties, and applications of CDFs, as well as how to choose the right CDF for your application.