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Understanding the exponential function is key when dealing with probability density functions (pdf). In its simplest form, the exponential function is written as \(e^x\). It is a mathematical function that grows rapidly with increasing x, making it a powerful tool to describe growth processes.

The function in the original problem is \(f(x) = e^{-x/2}\), a modified version of the exponential function. Here, the exponent has been scaled by -1/2, which influences the function's behavior:

• The exponential function is always positive. Since the exponent is negative, the function will decrease as x increases.
• This characteristic makes it suitable for modeling exponential decay, common in probability contexts, to represent decreasing likelihood.

When interpreting \(e^{-x/2}\), it is crucial to remember these points. It ensures the function remains positive for all real x, including the specified interval \([0, \ln(4)]\). This positivity is a requirement for defining a valid pdf in probabilistic terms.

The integral calculation is the process to find the total area under a curve, which is essential for determining the validity of a pdf over a specified interval.

The aim is to calculate the integral of \(f(x) = e^{-x / 2}\) from 0 to \(\ln(4)\). This involves substitution and integration by parts. Here's a simplified overview:

• Introduce a substitution, such as \(u = -x/2\), to simplify the integral.
• This transforms the integral to \(-2\int e^u \, du\), where -2 is the factor from the derivative of \( -x/2 \).
• The solution of the integral becomes: \(-2[e^{u}]_0^{\ln(2)} = -2[e^{-x/2}]_0^{\ln(2)}\).

Solve this to compute the definite integral.

• Evaluate at upper limit: \(-2(e^{-\ln(2)/2}) = -2 (1/2) = -1\).
• Evaluate at lower limit: \(-2(e^{0}) = -2 \times 1 = -2\).
• Subtract the results: \(-2 - (-1) = -1\).

This calculation error implies the probabilities don't sum to 1, critiquing the function's given form for pdf validity.

Ensuring function positivity is the first step in determining a valid probability density function. Positivity means that the function should not take any negative values. This requirement is crucial because probabilities themselves cannot be negative.

Let's break down the positivity of the given exponential function \(f(x) = e^{-x / 2}\):

• Exponential functions \(e^x\) are inherently positive, no matter what the exponent is, meaning any transformation still yields a positive value.
• In the scenario where the exponent, \(-x/2\), is negative, the value of the function decreases but stays above zero.
• This ensures that the function is positive over the interval \([0, \ln(4)]\) and beyond.

Function positivity is fundamental in the practical application of pdfs. While this function satisfies the positivity requirement, the negative area found in integration indicates a different problem, hinting at possible errors in the setup or chosen interval for this pdf. Correct setup and calculations are necessary to satisfy both positivity and total area equating to one for a true probability density function.