Problem 15 - Calculus for Biology and Medicine

Question

Assume that \(X\) is exponentially distributed with parameter \(\lambda=3.0\). (a) Assume that a sample of size 50 is taken from this population. What is the approximate distribution of the sample mean? (b) Assume now that 1000 samples, each of size 50 , are taken from this population and a histogram of the sample means of each of the samples is produced. What shape will the histogram be approximately?

Step-by-Step Solution

Answer

(a) Approximately normal with mean \(\frac{1}{3}\) and variance \(\frac{1}{450}\). (b) The histogram will be approximately normal.

Step 1: Recognize the Distribution

An exponentially distributed random variable has a probability distribution given by the parameter \(\lambda\). Here, \(X\) follows an exponential distribution with \(\lambda = 3.0\). This means that the mean of the distribution is \(\frac{1}{\lambda} = \frac{1}{3}\) and the variance is \(\frac{1}{\lambda^2} = \frac{1}{9}\).

Step 2: Determine the Distribution of the Sample Mean

When taking a sample of size \(n=50\) from an exponential distribution with mean \(\frac{1}{3}\), the sample mean \(\overline{X}\) will have an approximately normal distribution due to the Central Limit Theorem. The mean of the sample mean is \(\mu = \frac{1}{3}\) and the variance is \(\frac{\sigma^2}{n} = \frac{1/9}{50} = \frac{1}{450}\).

Step 3: Central Limit Theorem

According to the Central Limit Theorem, for large sample sizes (\(n=50\) is considered sufficiently large), the distribution of the sample mean \(\overline{X}\) approaches a normal distribution. Thus, \(\overline{X} \sim N\left(\frac{1}{3}, \frac{1}{450}\right)\), which means it is approximately normally distributed with mean \(\frac{1}{3}\) and variance \(\frac{1}{450}\).

Step 4: Distribution Shape for Histogram

When 1000 samples, each of size 50, are taken from the population and the sample means are plotted in a histogram, the Central Limit Theorem suggests the shape of the histogram will be approximately normal. This is due to the sample means being normally distributed as established in the previous step.

Key Concepts

Exponential Distribution

An exponential distribution is a continuous probability distribution used to model the time or space between events in a Poisson process. In a Poisson process, events occur continuously and independently at a constant average rate. The exponential distribution is characterized by its parameter \( \lambda \), known as the rate parameter. For example, if \( X \) is exponentially distributed with \( \lambda = 3.0 \), it signifies that the average rate of occurrence is 3 events per unit time.
The probability density function (PDF) of an exponential distribution is given by:

• \( f(x; \lambda) = \lambda e^{-\lambda x} \) for \( x \geq 0 \)

This distribution has a mean (expected value) of \( \frac{1}{\lambda} \) and a variance of \( \frac{1}{\lambda^2} \). In our example with \( \lambda = 3.0 \), the mean is \( \frac{1}{3} \approx 0.333 \) and the variance is \( \frac{1}{9} \approx 0.111 \).
The exponential distribution is memoryless, meaning that future probabilities do not depend at all on any past outcomes. This property makes it unique and particularly useful in various fields such as queueing theory, reliability engineering, and survival analysis.

Normal Distribution

The normal distribution, often referred to as the Gaussian distribution, is a continuous probability distribution that is symmetrical around its mean. Most of the observations cluster around a central peak, decreasing as you move away from the center. It is characterized by two parameters: the mean \( \mu \) and the standard deviation \( \sigma \).
The probability density function of a normal distribution is:

• \( f(x; \mu, \sigma^2) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)

In the context of the given exercise, the sample mean \(\overline{X}\) from samples of size 50 drawn from an exponential distribution converges in distribution to a normal distribution due to the Central Limit Theorem. Specifically, the mean of \( \overline{X} \) is \( \mu = \frac{1}{3} \) and its variance is \( \frac{1}{450} \). Therefore, \( \overline{X} \) can be approximated by \( N\left(\frac{1}{3}, \frac{1}{450}\right) \).
The significance of the normal distribution in statistics is due to the role it plays in the Central Limit Theorem, which implies that the means of a large number of repeated samples will be normally distributed regardless of the shape of the original data distribution.

Sample Mean

The sample mean is an important random variable derived from a random sample. It provides an estimate of the population mean. For a given set of data points, the sample mean \( \overline{X} \) is calculated as:

• \( \overline{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \)

where \( X_i \) refers to each individual observation in the sample, and \( n \) is the number of observations. In statistical analysis, particularly in inferential statistics, the sample mean is used to make inferences about the population mean.
In the exercise, the Central Limit Theorem plays a pivotal role in determining that the sampling distribution of the sample mean will approximate a normal distribution when the sample size is large enough, even if the underlying population distribution is not normal. In this case, sample means from an exponential distribution converge towards a normal distribution as the sample size (in each of the 1000 samples) grows to 50, leading to a normal-shaped histogram when plotting these means.
The sample mean is therefore a powerful tool in statistics, enabling analysts to summarize data, make predictions, and form the basis for statistical testing and confidence intervals.