Question

. Suppose \(X\) is a random variable with the pdf \(f(x)\) which is symmetric about \(0,(f(-x)=f(x))\). Show that \(F(-x)=1-F(x)\), for all \(x\) in the support of \(X\).

Short answer

The statement \(F(-x) = 1 - F(x)\) holds true in the case of symmetric pdfs because the probability that \(X\) takes a value greater than \(x\) is the same as the probability that it takes a value less than \(-x\).

Step-by-step solution

Define the Cumulative Distribution Function (CDF)

The CDF, \(F(x)\), of a random variable \(X\) is given by \(F(x) = \int_{-\infty}^{x}f(t) dt\). In this case, \(f(t)\) is the pdf of \(X\).

Compute \(F(-x)\) using the symmetry property

From the symmetry property \(f(t) = f(-t)\), we know that the CDF at \(-x\) is given by \(F(-x) = \int_{-\infty}^{-x}f(t) dt\). We can change the limit of integration from \(-\infty\) to \(+\infty\) and the variable of integration from \(t\) to \(-t\). This gives us \(F(-x) = \int_{\infty}^{x} f(-t) dt\). In view of the given symmetric property, \(f(-t) = f(t)\), this simplifies further to \(F(-x) = \int_{\infty}^{x} f(t) dt\).

Interpret \(1 - F(x)\)

The term \(1 - F(x)\) is the probability that the random variable \(X\) takes a value greater than \(x\). This is equivalent to the integral of the pdf from \(x\) to \(+\infty\), which is \(1 - F(x) = \int_{x}^{\infty} f(t) dt\). Since we showed in step 2 that \(F(-x) = \int_{\infty}^{x} f(t) dt\), it follows that \(F(-x) = 1 - F(x)\).

Conclusion

From steps 2 and 3, it is evident that \(F(-x) = 1 - F(x)\), as required, because the probability that \(X\) takes a value greater than \(x\) is equal to the probability that it takes a value less than \(-x\), due to the symmetry of the pdf about 0.

Key concepts

Probability Density Function

A Probability Density Function (PDF) is a crucial concept in understanding random variables in statistics and probability. When we think about a PDF, we're considering a function that describes the likelihood of different outcomes in a continuous random distribution. The PDF, denoted as \(f(x)\), helps us comprehend how probability is distributed over the values of the random variable, \(X\).

• The PDF is a non-negative function, meaning \(f(x) \geq 0\) for all \(x\).
• The integral of the PDF over the entire space is equal to 1. This is because the total probability of any random variable occurring must be 100% or \(\int_{-\infty}^{\infty} f(x) dx = 1\).
• Unlike probability mass functions for discrete variables, the value of the PDF at any single point \(x\) is not the probability. Instead, it is the relative likelihood and helps in calculating probabilities over an interval.

In our exercise, we know that the PDF is symmetric, meaning \(f(-x) = f(x)\), indicating that the distribution is mirrored perfectly around 0.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) is another pivotal tool when dealing with random variables. The CDF, represented as \(F(x)\), provides a running total of probabilities by indicating the probability that a random variable \(X\) will take a value less than or equal to \(x\).

• Mathematically, it is expressed as \(F(x) = \int_{-\infty}^{x} f(t) dt\), where \(f(t)\) is the PDF.
• The CDF is a non-decreasing function, meaning as \(x\) increases, \(F(x)\) either increases or stays the same.
• It ranges from 0 to 1, with \(\lim_{x \to -\infty} F(x) = 0\) and \(\lim_{x \to \infty} F(x) = 1\).

In our exercise, we demonstrated that for a symmetric PDF, the CDF satisfies an additional symmetry: \(F(-x) = 1 - F(x)\). This illustrates that for every point \(x\), the probability of being less than \(-x\) complements the probability of being greater than \(x\), thanks to the symmetry about 0.

Random Variable Properties

Random variables are foundational in the study of probability and statistics, representing quantities whose outcomes are determined by chance. In the context of the exercise, it's important to recognize a few key properties:

• Symmetry: The given random variable's PDF is symmetric about 0, implying that the random variable has an equal probability of being a certain distance on either side of the mean. Symmetry simplifies many calculations.
• Continuous: The variable in question is continuous, meaning it can take on an infinite number of values within a given range. This is reflected in the use of integrals when dealing with PDFs and CDFs.
• Support: The support of a random variable is the set of all values for which the PDF is non-zero. For a symmetric PDF around 0, this implies the support is symmetric as well, extending equally in both the negative and positive directions.

Understanding these properties helps clarify how the PDF and CDF behave under the conditions set by the exercise. Knowing that a function is symmetric, continuous, and has a specific support helps in predicting and verifying the behavior of functions related to the random variable.