Question

A function is a probability density function if it satisfies the following definition: \(\int_{-\infty}^{\infty} f(t) d t=1\). The probability that a random variable \(x\) lies between \(a\) and \(b\) is given by \(P(a \leq x \leq b)=\int_{a}^{b} f(t) d t\)Show that \(f(x)=\left\\{\begin{array}{c}0 \text { if } x<0 \\ 7 e^{-7 x} \text { if } x \geq 0\end{array}\right.\) is a probability density function.

Short answer

The function satisfies both conditions of a PDF: \( f(x) \geq 0 \) and the integral equals 1.

Step-by-step solution

Identify requirements for a probability density function

A function \( f(x) \) is a probability density function if it satisfies two conditions. First, \( f(x) \geq 0 \) for all \( x \). Second, the total area under the curve of \( f(x) \) across the entire space must equal 1, i.e., \( \int_{-\infty}^{\infty} f(t) \, dt = 1 \).

Check positivity of the function

The given function is \( f(x) = \begin{cases} 0, & \text{if } x < 0 \ 7e^{-7x}, & \text{if } x \geq 0 \end{cases} \). For \( x < 0 \), \( f(x) = 0 \). For \( x \geq 0 \), \( 7e^{-7x} \geq 0 \) because \( e^{-7x} \) is always positive. Thus, \( f(x) \geq 0 \) for all \( x \).

Evaluate the integral over the entire space

Since \( f(x) = 0 \) for \( x < 0 \), the integral from \(-\infty\) to \( 0 \) is 0. Now evaluate from \( 0 \) to \( +\infty \): \( \int_{0}^{\infty} 7e^{-7x} \, dx \).

Compute the integral for the positive domain

Substitute into the integral: \( \int_{0}^{\infty} 7e^{-7x} \, dx = 7 \int_{0}^{\infty} e^{-7x} \, dx \). Use the substitution \( u = -7x \), \( du = -7dx \) or \( dx = -\frac{1}{7} du \). Thus the integral becomes \( -1 \int_{0}^{-\infty} e^{u} \, du \).

Evaluate the new integral

Now evaluate the integral \( - \int_{0}^{-\infty} e^{u} \, du = -\left[ e^{u} \right]_{0}^{-\infty} = -[0 - 1] = 1 \). Hence, \( \int_{0}^{\infty} 7e^{-7x} \, dx = 1 \).

Confirm that the integral equals 1

The integral from Step 5 confirms \( \int_{0}^{\infty} 7e^{-7x} \, dx = 1 \). Thus, \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \), fulfilling both conditions for a probability density function.

Key concepts

Integral Calculus

Integral calculus is a branch of calculus concerned with the concept of integrals and their applications. An integral can be visualized as the area under a curve, which is crucial for analyzing functions in various fields, including probability.
In the context of probability density functions (PDFs), integral calculus helps us find the total probability across the entire spectrum of possible outcomes.

• The indefinite integral is a function representing the antiderivative of another function.
• The definite integral, which is more relevant here, involves two endpoints and calculates the area under the curve between them.

For a function to qualify as a probability density function, the integral of the function over all possible values must be 1. This integral sums up the entire probability "mass," assuring it accounts for 100% of the outcomes.
By using integral calculus, we ensure the function satisfies the core principle of total probability equaling 1.

Exponential Functions

Exponential functions play a vital role in various branches of mathematics and science. They are defined by expressions in the form of \( f(x) = a e^{bx} \), where \( e \) is Euler's number, an important constant approximately equal to 2.71828.
In probability theory, exponential functions often describe processes related to growth or decay, such as radioactive decay or interest compounding. They are particularly useful in probability density functions for continuous random variables.

• An exponential function is characterized by its rapid increase (or decrease) and constant ratio of growth.
• It inherently models situations of constant percentage change over equal intervals.

For example, in the given PDF, the function is \( f(x) = 7e^{-7x} \) for \( x \geq 0 \). This form ensures that the function has a rapidly decaying nature, which is common for distributions related to time between events, like in the exponential distribution used in queuing theory.
Understanding exponential functions allows us to manipulate and integrate them properly, thus confirming their properties as probability functions.

Probability Theory

Probability theory is the mathematical framework for quantifying uncertainty and provides tools to analyze random events. It allows us to model real-world phenomena where outcomes are inherently unpredictable but follow certain patterns or distributions.
In the context of probability density functions, probability theory provides the foundational rules to ensure that outcomes behave as expected. Here are the two major requirements for a PDF:

• The function must be non-negative for all values, reflecting the fact that probabilities cannot be negative.
• The total probability across all possible outcomes must equal 1. This ensures completeness; all possible scenarios are accounted for.

The function from the exercise, \( f(x) \), meets these criteria, as it is non-negative for all \( x \) and its integral over its domain equals 1.
By understanding probability theory, we can approach problems with a structured way of verifying functions within the probabilistic models we've set, such as demonstrating whether a given function can be used as a PDF for modeling real-world phenomena.