Problem 10 - Algebra 2

Question

For Exercises \(9-14\) , use the following information. An exponential distribution has a mean of \(0.5 .\) Find each probability. $$ x>3 $$

Step-by-Step Solution

Answer

The probability \( P(x > 3) \) is approximately 0.0025.

Step 1: Understand the Exponential Distribution

An exponential distribution is a continuous probability distribution used to model the time between events in a Poisson process. It is characterized by its mean, denoted as \( \mu \). For this problem, \( \mu = 0.5 \).

Step 2: Identify the Parameter \( \lambda \)

The rate parameter \( \lambda \) of an exponential distribution is the reciprocal of the mean, i.e., \( \lambda = \frac{1}{\mu} \). Given \( \mu = 0.5 \), we have \( \lambda = \frac{1}{0.5} = 2 \).

Step 3: Write the Exponential Distribution Formula

The probability density function (PDF) of an exponential distribution is given by:\[P(x > x_0) = e^{-\lambda x_0}\]We will use this formula to calculate the probability.

Step 4: Plug in Values to Find the Probability

We are required to find \( P(x > 3) \). Using the PDF:\[P(x > 3) = e^{-2 \cdot 3} = e^{-6}\]

Step 5: Calculate the Numerical Probability

Using a calculator, evaluate \( e^{-6} \). Thus,\[P(x > 3) \approx 0.0024787521766663585\]This is the probability that \( x \) is greater than 3.

Key Concepts

Probability

Probability helps us describe uncertainty or the chance of an event occurring. It is measured between 0 and 1, where 0 means an event will not occur, and 1 means it will certainly occur. In the context of exponential distribution, probability is used to determine how likely it is for an event to occur after a certain time has passed.

In our exercise, we are looking for the probability that a time duration is greater than 3 units. This is calculated using the exponential distribution's probability density function, which is tailored for events in a continuous setting.

Some key points regarding probability in an exponential distribution are:

• Since it is a continuous distribution, probabilities are concerned with time intervals rather than discrete outcomes.
• The cumulative probability decreases as the time interval increases, which means the likelihood of longer waiting times is lower.

Understanding probability is crucial for making predictions and informed decisions regarding time-based events.

Probability Density Function

A Probability Density Function (PDF) describes the likelihood of a random variable to take on a particular value in a continuous probability distribution. In the exponential distribution, the PDF helps us understand the distribution of times between events in a Poisson process, such as waiting times or time intervals.

For an exponential distribution with a rate parameter \( \lambda \), the PDF is formulated as:

• \( f(x) = \lambda e^{-\lambda x} \) for \( x \geq 0 \)
• This function is only defined for non-negative values, reflecting the reality that time cannot be negative.

The PDF is integral in solving exercises like finding \( P(x > 3) \) because it allows us to calculate probabilities over an interval using the formula \( P(x > x_0) = e^{-\lambda x_0} \).

In essence, the PDF is utilized to determine how rapidly or slowly events occur over time in a Poisson process. It plays a vital role in characterizing the exponential distribution and making sense of event timings.

Poisson Process

The Poisson process is a statistical model used for describing random events that occur independently and sporadically in time or space. It is particularly useful for modeling events that happen over time, such as phone calls at a call center or emails received in an inbox.

A few key characteristics of a Poisson process include:

• Events occur independently of each other.
• The average rate at which events occur remains constant over time.
• Two events cannot occur at the exact same time.

In connection with the exponential distribution, the Poisson process helps describe the timing of events. The time between two events in a Poisson process is exponentially distributed. This direct relation allows us to estimate waiting times or time intervals between occurrences using the PDF of an exponential distribution.

The Poisson process, therefore, serves as the backbone for understanding and applying the concept of exponential distribution in practical scenarios where time intervals between events are of interest.