Properties of the Probability Density Function

Oct 13, 2025 by windailys.com 44 views

Introduction

In probability theory, a probability density function (PDF) is a fundamental concept used to describe the distribution of a random variable. It provides a way to calculate the probability of a random variable taking on a value within a given range. In this article, we will explore the probability density function of a random variable $Y$ and discuss its properties.

The Probability Density Function of $Y$

The probability density function of $Y$ is given by:

f(y) = \begin{cases} \frac{1}{4}(3-y), & \text{if } 0 \lt y \lt 2 \\ 0, & \text{otherwise.} \end{cases}

This function is defined for $y$ values between 0 and 2, and it is zero outside this range. The function is also continuous and non-negative, which are essential properties of a probability density function.

Properties of the Probability Density Function

To understand the properties of the probability density function of $Y$ , let's analyze its behavior within the defined range.

Non-Negativity

The probability density function of $Y$ is non-negative for all $y$ values between 0 and 2. This means that the probability of $Y$ taking on a value within this range is always positive.

Continuity

The probability density function of $Y$ is continuous within the defined range. This means that the function can be expressed as a continuous mathematical expression, and it does not have any discontinuities.

Normalization

To ensure that the probability density function of $Y$ is properly normalized, we need to verify that the integral of the function over the entire range is equal to 1.

\int_{0}^{2} \frac{1}{4}(3-y) dy = \left[ \frac{1}{4}(3y - \frac{y^2}{2}) \right]_{0}^{2} = \frac{1}{4}(6 - 2) = 1

This confirms that the probability density function of $Y$ is properly normalized.

Calculating Probabilities

With the probability density function of $Y$ in hand, we can now calculate probabilities of $Y$ taking on values within a given range.

Probability of $Y$ Taking on a Value within a Given Range

To calculate the probability of $Y$ taking on a value within a given range, we need to integrate the probability density function over the specified range.

For example, let's calculate the probability of $Y$ taking on a value between 0 and 1.

P(0 \lt Y \lt 1) = \int_{0}^{1} \frac{1}{4}(3-y) dy = \left[ \frac{1}{4}(3y - \frac{y^2}{2}) \right]_{0}^{1} = \frac{1}{4}(3 - \frac{1}{2}) = \frac{5}{8}

This means that the probability of $Y$ taking on a value between 0 and 1 is $\frac{5}{8}$ .

Probability of $Y$ Taking on a Specific Value

To calculate the probability of $Y$ taking on a specific value, we need to evaluate the probability density function at that value.

For example, let's calculate the probability of $Y$ taking on the value 1.

P(Y = 1) = f(1) = \frac{1}{4}(3-1) = \frac{1}{2}

This means that the probability of $Y$ taking on the value 1 is $\frac{1}{2}$ .

Conclusion

In this article, we explored the probability density function of a random variable $Y$ and discussed its properties. We verified that the function is non-negative, continuous, and properly normalized. We also calculated probabilities of $Y$ taking on values within a given range and specific values. This provides a solid foundation for further analysis and applications of the probability density function of $Y$ .

References

Jaynes, E.T. (2003). Probability Theory: The Logic of Science. Cambridge University Press.
Ross, S.M. (2012). Introduction to Probability Models. Academic Press.
Henley, J. (2013). Probability and Statistics. Cengage Learning.
Frequently Asked Questions about the Probability Density Function of $Y$ ====================================================================

Q: What is the probability density function of $Y$ ?

A: The probability density function of $Y$ is given by:

f(y) = \begin{cases} \frac{1}{4}(3-y), & \text{if } 0 \lt y \lt 2 \\ 0, & \text{otherwise.} \end{cases}

Q: What are the properties of the probability density function of $Y$ ?

A: The probability density function of $Y$ is non-negative, continuous, and properly normalized. This means that the function is always positive, can be expressed as a continuous mathematical expression, and the integral of the function over the entire range is equal to 1.

Q: How do I calculate the probability of $Y$ taking on a value within a given range?

A: To calculate the probability of $Y$ taking on a value within a given range, you need to integrate the probability density function over the specified range. For example, let's calculate the probability of $Y$ taking on a value between 0 and 1.

P(0 \lt Y \lt 1) = \int_{0}^{1} \frac{1}{4}(3-y) dy = \left[ \frac{1}{4}(3y - \frac{y^2}{2}) \right]_{0}^{1} = \frac{1}{4}(3 - \frac{1}{2}) = \frac{5}{8}

Q: How do I calculate the probability of $Y$ taking on a specific value?

A: To calculate the probability of $Y$ taking on a specific value, you need to evaluate the probability density function at that value. For example, let's calculate the probability of $Y$ taking on the value 1.

P(Y = 1) = f(1) = \frac{1}{4}(3-1) = \frac{1}{2}

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

A: The expected value of $Y$ is given by:

E[Y] = \int_{0}^{2} y \cdot f(y) dy = \int_{0}^{2} y \cdot \frac{1}{4}(3-y) dy = \left[ \frac{1}{4} \left( \frac{3y^2}{2} - \frac{y^3}{3} \right) \right]_{0}^{2} = \frac{1}{4} \left( 6 - \frac{8}{3} \right) = \frac{10}{12} = \frac{5}{6}

Q: What is the variance of $Y$ ?

A: The variance of $Y$ is given by:

Var[Y] = E[Y^2] - (E[Y])^2 = \int_{0}^{2} y^2 \cdot f(y) dy - (E[Y])^2 = \int_{0}^{2} y^2 \cdot \frac{1}{4}(3-y) dy - \left( \frac{5}{6} \right)^2 = \frac{1}{4} \left( \frac{3y^3}{3} - \frac{y^4}{4} \right)_{0}^{2} - \frac{25}{36} = \frac{1}{4} \left( 8 - 4 \right) - \frac{25}{36} = \frac{4}{4} - \frac{25}{36} = \frac{36}{36} - \frac{25}{36} = \frac{11}{36}

Q: Can I use the probability density function of $Y$ to model real-world phenomena?

A: Yes, the probability density function of $Y$ can be used to model real-world phenomena. For example, you can use it to model the distribution of heights of a population, the distribution of exam scores, or the distribution of waiting times in a queue.

Q: How do I apply the probability density function of $Y$ in practice?

A: To apply the probability density function of $Y$ in practice, you need to:

Identify the problem you want to solve.
Determine the relevant parameters of the problem.
Use the probability density function of $Y$ to model the problem.
Calculate the desired probabilities or expected values using the probability density function of $Y$ .
Interpret the results in the context of the problem.

By following these steps, you can effectively apply the probability density function of $Y$ to solve real-world problems.