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An exponential random variable describes the time between events in a Poisson process. These events occur continuously and independently at a constant average rate. Examples include the time until a radioactive particle decays, the time between bus arrivals, or in this case, the time between successive U.S. Supreme Court resignations.

The exponential distribution has a single parameter \(\lambda\), the rate parameter, which is the reciprocal of the expected value (mean). For a random variable \(X\) with exponential distribution and expected value 2 years, the rate parameter \(\lambda\) is computed as follows:

\[ \lambda = \frac{1}{EX} = \frac{1}{2} = 0.5 \]

Understanding this parameter helps determine probabilities of varying durations for the time between resignations.

The expected value of a random variable provides a measure of the center of the distribution. For an exponential distribution, it is important since the rate parameter \(\lambda\), and the expected value are closely related. Given an expected value of 2 years for the time between Supreme Court resignations, we can derive the rate parameter to be 0.5, as seen before.

Mathematically, the relationship between the expected value and \(\lambda\) is expressed as:

• \[ E(X) = \frac{1}{\lambda} \]
• \[ \lambda = \frac{1}{E(X)} \]

The expected value, therefore, allows us to fully characterize the exponential distribution with a single parameter. This makes it easier to understand and apply in real-world scenarios, like predicting the time between Supreme Court resignations.

Calculating probabilities in an exponential distribution often revolves around finding how likely it is that the time between events exceeds or is less than a certain value. For our scenario, we need to determine the probability that the Supreme Court composition remains unchanged for 5 years or more.

The general formula for finding the probability that an exponential random variable surpasses a value \(x\) is:

\[ P(X > x) = e^{-\lambda x} \]

Given the expected value of 2 years (hence \(\lambda = 0.5\)), and that we want to find the probability of no resignations for at least 5 years, we plug in the values:

\[ P(X > 5) = e^{-0.5 \times 5} = e^{-2.5} \]

Using a calculator, we get:

\[ e^{-2.5} \approx 0.0821 \]

This means there is approximately an 8.21% chance that there will be no changes in the Supreme Court over a period of 5 years.