Mathematics (9660) | Exponential distribution

📚 Exponential Distribution: Continuous Waiting Times (S2.3)

Welcome to the chapter on the Exponential Distribution! This is one of the most practical continuous distributions you will encounter in A-Level Statistics. While the Poisson distribution helped us count how many events happened in a fixed time, the Exponential distribution helps us model the time or distance *between* those events.

Think of it as the mathematical description of "waiting time." Understanding this distribution is crucial for real-world applications like modelling reliability, queues, and radioactive decay.

---

1. Defining the Exponential Distribution

1.1 What is a Continuous Random Variable? (Quick Recap)

Unlike discrete variables (like the Binomial or Poisson) which can only take specific integer values, a Continuous Random Variable, \(X\), can take any value within a given range. Since there are infinite possibilities, we cannot assign a probability to a single point, so we use functions:

• Probability Density Function (PDF), \(f(x)\): Used for calculus (integration) to find probabilities over an interval.
• Cumulative Distribution Function (CDF), \(F(x)\): Used to find the probability that \(X\) is less than or equal to a certain value.

Remember: For any continuous variable \(X\), the probability of hitting an exact value is zero: \(P(X = a) = 0\). This means \(P(X < a)\) is the same as \(P(X \le a)\).

1.2 The Formulas You Need to Know

The Exponential distribution is defined by a single parameter, \(\lambda\) (lambda), which represents the rate of events.

A. Probability Density Function (PDF)

The PDF for an Exponential variable \(X\) is:

$$f(x) = \lambda e^{-\lambda x} \quad \text{for } x \ge 0$$

Note: The variable \(X\) must be non-negative because you cannot wait a negative amount of time!

B. Cumulative Distribution Function (CDF)

The CDF is the integral of the PDF from 0 to \(x\). This formula is your shortcut for calculating probabilities:

$$F(x) = P(X \le x) = 1 - e^{-\lambda x}$$

Key Takeaway: The parameter \(\lambda\) dictates the shape of the distribution, and the CDF is the easiest way to calculate most probabilities.

---

2. Conditions for Application (The Link to Poisson)

The exponential distribution doesn't just appear randomly; it has a direct relationship with the Poisson distribution.

The exponential distribution is used to model the length of intervals between events that occur according to a Poisson process.

If events happen randomly and independently at a constant average rate \(\lambda\) (a Poisson process), then:

1. The number of events in a fixed interval follows a Poisson distribution with parameter \(\lambda\).

2. The time/distance between those events follows an Exponential distribution with parameter \(\lambda\).

Example Analogy: Waiting for a Bus

Suppose buses arrive at a rate of 4 per hour.

• Poisson: How many buses arrive in the next 2 hours? (\(\lambda_{Poisson} = 4 \times 2 = 8\))
• Exponential: How long will you wait for the next bus? (\(\lambda_{Exp} = 4\))

The key is that the rate \(\lambda\) is the same in both models, but for the Exponential distribution, it must be the rate in terms of events per unit of time/distance.

Did you know? The Exponential distribution is the only continuous distribution with the memoryless property. This means that if you are waiting for an event, the probability that you will wait an extra 5 minutes does not depend on how long you have already waited. The distribution effectively "forgets" history.

Key Takeaway: Use the Exponential distribution whenever you are measuring the continuous time (or space) until the *first* event, or the time *between* sequential events, in a process governed by a constant rate \(\lambda\).

---

3. Calculating Probabilities Step-by-Step

Calculating probabilities involves using the CDF, \(F(x) = 1 - e^{-\lambda x}\), or working with its complement.

Let \(X\) be the random variable representing the waiting time.

3.1 The Three Main Probability Types

Case 1: Probability Less Than (\(P(X \le a)\))

This is the most straightforward calculation. Use the CDF directly.

$$P(X \le a) = F(a) = 1 - e^{-\lambda a}$$

Example: If \(\lambda = 0.5\), find the probability the waiting time is less than 3 (\(P(X \le 3)\)).

$$P(X \le 3) = 1 - e^{-(0.5)(3)} = 1 - e^{-1.5} \approx 0.7769$$

Case 2: Probability Greater Than (\(P(X > a)\))

Since the CDF gives \(P(X \le a)\), we find the probability of waiting longer than \(a\) using the complement rule:

$$P(X > a) = 1 - P(X \le a) = 1 - (1 - e^{-\lambda a})$$

This simplifies beautifully to:

$$P(X > a) = e^{-\lambda a}$$

Example: If \(\lambda = 0.5\), find the probability the waiting time is more than 5 (\(P(X > 5)\)).

$$P(X > 5) = e^{-(0.5)(5)} = e^{-2.5} \approx 0.0821$$

Case 3: Probability Between Two Values (\(P(a < X < b)\))

Use the rule for continuous variables: \(P(a < X < b) = F(b) - F(a)\).

$$P(a < X < b) = (1 - e^{-\lambda b}) - (1 - e^{-\lambda a})$$

This simplifies to:

$$P(a < X < b) = e^{-\lambda a} - e^{-\lambda b}$$

Common Mistake to Avoid: Never forget that the parameter \(\lambda\) is the rate. Ensure your time units (hours, minutes, days) match the units used to define \(\lambda\). If \(\lambda\) is 0.5 per minute, all time values (x, a, b) must be in minutes.

3.2 Using Integration (When CDF is Not Used)

The syllabus allows for calculating probabilities by integrating the PDF. While usually much slower than using the CDF, you must know the principle:

$$P(a < X < b) = \int_{a}^{b} f(x) dx = \int_{a}^{b} \lambda e^{-\lambda x} dx$$

Don't worry if this seems tricky at first; remember that the integral of \( \lambda e^{-\lambda x} \) is simply \( -e^{-\lambda x} \). Applying the limits \(a\) and \(b\) gives you the same result as the simplified CDF formula above!

Key Takeaway: Always use the CDF shortcuts: \(P(X \le a) = 1 - e^{-\lambda a}\) and \(P(X > a) = e^{-\lambda a}\). This saves crucial time in exams.

---

4. Mean, Variance, and Standard Deviation

For an Exponential distribution, the theoretical mean and variance are derived through integration (but the syllabus confirms you do not need to prove them—just know the results!).

If \(X \sim \text{Exp}(\lambda)\):

4.1 Mean (Expected Value)

The expected waiting time, \(E(X)\) or \(\mu\), is the reciprocal of the rate \(\lambda\).

$$E(X) = \mu = \frac{1}{\lambda}$$

Analogy: If buses arrive at a rate of \(\lambda = 4\) per hour, the average time you expect to wait is \(1/4\) of an hour, or 15 minutes.

4.2 Variance and Standard Deviation

The variance, \(Var(X)\) or \(\sigma^2\), and the standard deviation, \(\sigma\), are also simple functions of \(\lambda\).

$$\text{Variance: } Var(X) = \sigma^2 = \frac{1}{\lambda^2}$$

$$\text{Standard Deviation: } \sigma = \sqrt{\frac{1}{\lambda^2}} = \frac{1}{\lambda}$$

Example Calculation

Suppose the lifespan of a certain electronic component (in years) follows an Exponential distribution with \(\lambda = 0.2\).

• Mean Lifespan: \(E(X) = \frac{1}{0.2} = 5\) years.
• Variance: \(Var(X) = \frac{1}{0.2^2} = \frac{1}{0.04} = 25\).
• Standard Deviation: \(\sigma = \frac{1}{0.2} = 5\) years. (Matches the mean, as expected!)

Key Takeaway: The formulas for mean and standard deviation are \(1/\lambda\), and the variance is \(1/\lambda^2\). They are simple, but you must memorize them.

---

Chapter Summary & Final Tips

The Exponential Distribution is essential for modelling continuous waiting times in systems where events happen at a constant rate. Always connect it back to the Poisson distribution to confirm your choice of \(\lambda\).

Quick Checklist for Exam Success

1. Context Check: Are we measuring continuous time/distance until an event? (Yes? Use Exponential.)

2. Parameter \(\lambda\): What is the rate? (If the question says the mean time is 10 minutes, then \(\mu = 10\), so \(\lambda = 1/10 = 0.1\). Be careful!)

3. Probability Shortcuts:

• \(P(X \le a) = 1 - e^{-\lambda a}\) (CDF)
• \(P(X > a) = e^{-\lambda a}\) (Survival Function)

4. Moments:

• Mean \(E(X) = 1/\lambda\)
• Variance \(Var(X) = 1/\lambda^2\)

Keep practicing those probability calculations—especially \(P(X > a)\) which is often used in reliability questions! You’ve got this!