91Ó°ÊÓ

In probability theory, it's important for a probability density function (PDF) to be non-negative over its domain. This means that the function must return values that are zero or greater for every possible input. For the function given in the exercise, \( f(x) = 2^x \), this requirement is satisfied. How do we know? Because the exponential function \( 2^x \) is always positive. No matter the value of \( x \), as long as it's real, \( 2^x \) will result in a positive number. This non-negativity is a key feature of exponential functions, ensuring they can potentially serve as probability density functions. For instance:

• \( 2^0 = 1 \), which is positive
• \( 2^1 = 2 \), which is positive
• Even \( 2^{-1} = \frac{1}{2} \), which remains positive

This characteristic ensures that \( f(x) = 2^x \) is a candidate for a PDF over any interval where we're examining it.

A fundamental property of probability density functions is that their definite integral, over the entire space in which they are defined, must equal exactly 1. This total of "1" represents the certainty that some outcome within that space will occur. In our exercise, we compute the definite integral of the function \( f(x) = 2^x \) over the interval \( \left[0, \frac{\ln(1+\ln 2)}{\ln 2}\right] \). Performing this integral involves finding the antiderivative of \( 2^x \), which is \( \frac{2^x}{\ln 2} \). Once you've found the antiderivative, you evaluate it between the limits:

• Plug in the upper limit \( \frac{\ln(1+\ln 2)}{\ln 2} \)
• Then plug in the lower limit 0
• Subtract these results to find the definite integral's total

When calculated, this integral evaluates to 1, confirming that \( f(x) \), when integrated over its specified range, satisfies the main requirement of a probability density function.

The exponential function \( 2^x \) is a captivating and vital mathematical function with distinctive properties, especially useful in probability and statistics. Exponential functions are characterized by their constant rate of growth. The graph of an exponential function like \( 2^x \) is always curving upwards for positive \( x \), indicating increasing growth, and it reflects symmetry over the y-axis for negative \( x \). Some key features of the exponential function:

• Rapid growth: As \( x \) increases, \( 2^x \) rises sharply.
• Decay when \( x \) is negative: It approaches zero but never actually gets there.
• Non-zero for all \( x \): The function never touches the x-axis.

These characteristics make exponential functions an excellent choice for modelling real-world scenarios involving growth, decay, and distributions, such as populations or radioactive decay. Knowing how the exponential function behaves is crucial for understanding why \( 2^x \) can be utilitarian as a probability density function over certain intervals.