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The probability density function (PDF) is a fundamental concept when dealing with continuous random variables, such as those described by the exponential distribution. For an exponential distribution with a parameter \( \beta \), the PDF is given by:

• \( f(y) = \frac{1}{\beta} e^{-y/\beta} \text{ for } y \geq 0 \)

This formula tells us that the probability density of any specific point, \( y \), is determined by the rate parameter \( \beta \).
The parameter \( \beta \) is not only a scale parameter but also the mean of the distribution. The PDF decreases exponentially because of the term \( e^{-y/\beta} \), which indicates that larger values of \( y \) have a rapidly decreasing likelihood.
Understanding the PDF is essential because it helps us visualize how likely different outcomes are, even though, for continuous variables, the exact probability of any single outcome is technically zero. Instead, the PDF shows us the relative likelihood instead.

The cumulative distribution function (CDF) is vital for understanding how probabilities build up over a range of values. For the exponential distribution, the CDF is given by the formula:

• \( F(y) = 1 - e^{-y/\beta} \)

This function describes the collective probability that a random variable \( Y \) is less than or equal to some value \( y \). It builds upon the concept of the PDF by accumulating probabilities.
In practical terms, while the PDF gives us the density at a particular point, the CDF gives us the complete probability for a range of values from zero up to \( y \).
For example, when the CDF equals 0.5, we reach the median of the distribution, meaning there is a 50% chance that the random variable will take a value less than or equal to \( y \).
In the exponential setting, this value is found by solving \( 1 - e^{-y/\beta} = 0.5 \). This solution shows how central the idea of 'cumulative probability' is in determining key metrics like the median.

The median is a significant statistical measure that provides insight into the central tendency of a distribution. In the context of the exponential distribution, it represents the point at which half the probability mass of the distribution falls below and half falls above.
To find the median of an exponential distribution with mean \( \beta \), you set the CDF to 0.5 since the median is the value where there is an equal probability of falling below or above it.

• Solve: \( 1 - e^{-y/\beta} = 0.5 \)

By solving this equation, we find the median value \( y \), which is calculated as:

• \( y = -\beta \ln(0.5) \)

This value effectively splits the distribution into two equal halves of probability, both having 50% probability. Understanding the median in the context of the exponential distribution helps us grasp the 'typical' value around which data points congregate, despite the skewed nature of the distribution.