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Probability distributions are essential in statistics as they describe how the probabilities are spread out over the possible values of a random variable. A good understanding of probability distributions helps predict outcomes and make informed decisions.

In the context of exponential distribution, which is often used to model the time until an event occurs, the distribution is defined for a positive random variable \(x\). It is characterized by a single parameter \(\theta\), often referred to as the rate parameter. This distribution is crucial when modeling processes that occur continuously and independently at a constant average rate.

An exponential distribution's probability density function (PDF) is given by:

• \( f(x; \theta) = \frac{1}{\theta} e^{-x/\theta} \) for \( x \geq 0 \)

These components help us understand how likely different outcomes are, facilitating predictions regarding the time or location of future events.

The mean and standard deviation are vital concepts when exploring any distribution, including exponential distribution. They provide insights into the distribution's center and variability, respectively.

For an exponential distribution, the mean \( \mu \) and the standard deviation \( \sigma \) are both equal to the parameter \( \theta \). Therefore, if \( \theta = 3 \), as mentioned in our exercise, we calculate:

• Mean (\( \mu \)) \( = \theta = 3\)
• Standard deviation (\( \sigma \)) \( = \theta = 3\)

Both statistics indicate that the data points tend to cluster around the mean of 3 and spread out to the same degree as this mean.

Using these values, you can define intervals such as \( \mu \pm 2\sigma \), helping assess the range where a majority of values lie, effectively enhancing the understanding of variability in the distribution.

The cumulative distribution function (CDF) describes the probability that a random variable \( x \) will take a value less than or equal to \( x \). It is a crucial tool in probability for assessing cumulative probability. For an exponential distribution, the CDF is expressed as:

• \( F(x; \theta) = 1 - e^{-x/\theta} \)

This function helps determine probabilities over intervals. For instance, considering the exercise, to find the probability that \( x \) falls within the interval \([0, 9]\), we use the CDF.

Calculate:

• \( P(x \leq 9) = 1 - e^{-9/3} = 1 - e^{-3} \)
• \( P(x \leq 0) = 1 - e^{0} = 0 \)

As you compute, \( 1 - e^{-3} \approx 0.9502 \), indicating a 95% probability. This CDF provides comprehensive insight into the probability landscape of a random variable over specific intervals, aiding in effective decision-making based on these probabilities.