Problem 10 - Calculus for Biology and Medicine

Question

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are independent random variables with \(P\left(X_{i}>x\right)=e^{-2 x} .\) What can you say about \(\frac{1}{n} \sum_{i=1}^{n} X_{i}\) as \(n \rightarrow \infty ?\)

Step-by-Step Solution

Answer

The sample mean \( \frac{1}{n} \sum_{i=1}^{n} X_{i} \) converges to \( \frac{1}{2} \) as \( n \to \infty \).

Step 1: Identify Distribution of Xi

The given probability statement \( P(X_i > x) = e^{-2x} \) suggests that the variable \( X_i \) follows an exponential distribution. Based on this, identify the parameter of the exponential distribution. For an exponential random variable \( X \) with rate \( \lambda \), \( P(X > x) = e^{-\lambda x} \), which implies \( \lambda = 2 \). Therefore, \( X_i \sim \text{Exponential}(2) \).

Step 2: Calculate Expected Value and Variance

For an exponential distribution \( X \sim \text{Exponential}(\lambda) \), the mean (expected value) \( E[X] \) is \( \frac{1}{\lambda} \) and the variance \( \text{Var}(X) \) is \( \frac{1}{\lambda^2} \). With \( \lambda = 2 \), this gives \( E[X_i] = \frac{1}{2} \) and \( \text{Var}(X_i) = \frac{1}{4} \).

Step 3: Identify the Sample Mean's Behavior

We are interested in the behavior of the sample mean \( \frac{1}{n} \sum_{i=1}^{n} X_{i} \) as \( n \to \infty \). The expected value of the sample mean is \( E\left[\frac{1}{n} \sum_{i=1}^{n} X_i\right] = \frac{1}{n} \sum_{i=1}^{n} E[X_i] = \frac{1}{2} \).

Step 4: Use the Law of Large Numbers

According to the law of large numbers, the sample mean \( \frac{1}{n} \sum_{i=1}^{n} X_{i} \) will converge in probability to the expected value \( E[X_i] = \frac{1}{2} \) as \( n \to \infty \). This means that as the number of observations \( n \) increases, the sample mean will get arbitrarily close to \( \frac{1}{2} \).

Step 5: Conclusion: Behavior of the Sample Mean

As \( n \to \infty \), the sample mean \( \frac{1}{n} \sum_{i=1}^{n} X_{i} \) converges in probability to \( \frac{1}{2} \). This is a result of each \( X_i \) being an exponential random variable with mean \( \frac{1}{2} \) and by the application of the law of large numbers.

Key Concepts

Exponential Distribution

An exponential distribution is a continuous probability distribution used to model the time or space between events in a process where events occur continuously and independently at a constant rate.

• For a random variable following an Exponential distribution with rate parameter \( \lambda \), the probability that it takes a value greater than \( x \) is given by: \[ P(X > x) = e^{-\lambda x} \]
• This suggests the random variable \( X \) has an exponential distribution with rate \( \lambda \).

In the context of the problem, the exponential distribution's rate parameter is identified by comparing the provided equation, \( P(X_i > x) = e^{-2x} \), leading us to conclude that \( \lambda = 2 \). The exponential distribution is often used to model waiting times or time until occurrences, such as the time until a light bulb burns out or the time between phone calls at a call center.This distribution is memoryless, meaning that the probability of an event occurring in the future is independent of any past occurrences.

Expected Value

The expected value, often called the mean, of a random variable gives a measure of the central tendency or "average" of the probability distribution.

• For exponential distribution with rate \( \lambda \), the expected value \( E[X] \) is calculated as:\[ E[X] = \frac{1}{\lambda} \]
• In the given exercise, since \( \lambda = 2 \), the expected value of each \( X_i \) is:\[ E[X_i] = \frac{1}{2} \]

Knowing the expected value helps us to understand where the center of the distribution lies. In practical terms, for the scenario given in the exercise, it means that, on average, the value of \( X_i \) will be around \( 0.5 \).The expected value plays a crucial role when analyzing the long-term behavior of random variables and allows us to apply principles such as the Law of Large Numbers.

Variance

Variance is a key concept in probability and statistics, providing insight into the spread or dispersion of a set of values. It is calculated for a random variable to describe how much the values differ from the expected value.

• For an exponentially distributed random variable \( X \) with rate \( \lambda \), the variance is given by:\[ \text{Var}(X) = \frac{1}{\lambda^2} \]
• Given \( \lambda = 2 \) in our exercise, the variance of each \( X_i \) is:\[ \text{Var}(X_i) = \frac{1}{4} \]

The variance gives us a quantitative measure of how much individual observations in the dataset differ from the expected value. It tells us about the variability or volatility of the process being measured.For example, a smaller variance indicates that the observations tend to be very close to the expected value, whereas a larger variance suggests a broader distribution around the mean.Understanding variance is crucial for predicting future values and measuring risks associated with the distribution of the data.