Problem 25 - Algebra 2

Question

REASONING An exponential distribution function has a mean of 2. A fellow student says that the probability that \(x>2\) is \(0.5 .\) Determine whether this is sometimes, altways, or never true. Explain your reasoning.

Step-by-Step Solution

Answer

The statement is never true for this mean of 2.

Step 1: Recall the properties of the exponential distribution

The exponential distribution with a mean \( \mu \) has the probability density function (PDF) given by \( f(x) = \frac{1}{\mu} e^{-x/\mu} \) for \( x \geq 0 \). In this case, the mean \( \mu \) is given as 2.

Step 2: Calculate the cumulative distribution function (CDF)

The cumulative distribution function (CDF) of an exponential distribution for \( X > x \) is calculated as \( P(X > x) = 1 - F(x) \), where \( F(x) = 1 - e^{-x/\mu} \).

Step 3: Determine the specific CDF at x=2

Using the formula for the CDF, calculate \( P(X > 2) = 1 - (1 - e^{-2/2}) = e^{-1} \).

Step 4: Simplify and evaluate the expression

Calculate \( e^{-1} \), which approximates to 0.3679. Hence, \( P(X > 2) eq 0.5 \).

Step 5: Conclude the probability statement

Given that \( P(X > 2) \approx 0.3679 \), it doesn't equal 0.5 with a mean of 2. The statement is **never** true for this parameter.

Key Concepts

Probability Density Function

The probability density function (PDF) of an exponential distribution describes how the probabilities of different outcomes are distributed. For an exponential distribution with mean \( \mu \), the PDF is given by the formula:

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

This function graphs how likely it is for a particular outcome to occur. With the mean set to 2 in this exercise, we substitute into the formula:

• \( f(x) = \frac{1}{2} e^{-x/2} \)

The exponential distribution is particularly used to model the time between the occurrence of events that happen independently and continuously at a constant average rate. In simple terms, the PDF helps us identify the likelihood of certain lengths of time occurring.

Cumulative Distribution Function

The cumulative distribution function (CDF) helps us understand the probability that a random variable is less than or equal to a particular value. For the exponential distribution, the CDF is shown as:

• \( F(x) = 1 - e^{-x/\mu} \)

The CDF gives a running total of probabilities up to an endpoint \( x \). In the context of an exponential distribution with mean 2, this becomes:

• \( F(x) = 1 - e^{-x/2} \)

The CDF allows us to calculate the probability of events such as \( P(X > 2) \), which tells us whether a variable is more likely to be above a certain value. To find this, we use the complementary cumulative distribution, \( P(X > x) = 1 - F(x) \). This step is useful when considering probabilities over a range of values rather than specific points.

Mean of Distribution

The mean of an exponential distribution is a key parameter that influences its shape and behavior. In an exponential distribution, the mean, denoted by \( \mu \), determines how quickly the probabilities decay. A smaller mean leads to a steeper decay, while a larger mean spreads the distribution more widely over the possible values. For this exercise, we are given \( \mu = 2 \). This indicates that on average, the events described by this distribution occur in every 2 time units. It tells us that 2 is central to the distribution, but doesn’t mean that half of the events will occur before or after this time, as probabilities are calculated using the exponential formula. Understanding the mean helps us interpret what the distribution tells us about real-world processes, such as the duration between accidents or the time between arrivals.

Probability Calculation

Calculating probabilities with exponential distributions involves utilizing both the PDF and CDF. In this exercise, the focus is on determining \( P(X > 2) \) with a mean of 2. Using the equation for \( P(X > x) \), we derive as follows:

• \( P(X > 2) = 1 - F(2) = e^{-2/2} = e^{-1} \)
• This calculates to approximately 0.3679.

This result shows that the probability is roughly 36.79% that the value is greater than 2, given this exponential distribution. Through this exercise, we see why the initial statement "the probability that \( x>2 \) is 0.5" is incorrect. The distribution decays exponentially, meaning larger values become increasingly less likely, especially as you look beyond the mean.