Chapter 37: Central Limit Theorem

Learning Objectives

1. Calculate probability of a sample mean using a population mean and variance with unknown distribution
2. Use the Central Limit Theorem to construct the Normal approximation of the Binomial and Poisson distributions

The Central Limit Theorem

Theorem 1: Central Limit Theorem (CLT)

Let \(X_i\) be iid rv’s with common mean \(\mu\) and variance \(\sigma^2\), for \(i=1,2,\ldots,n\). Then \[\sum_{i=1}^n X_i \rightarrow \text{N}(n\mu, n\sigma^2)\]

Extension of the CLT

Corollary 1

Let \(X_i\) be iid rv’s with common mean \(\mu\) and variance \(\sigma^2\), for \(i=1,2,\ldots,n\). Then \[\overline{X}=\frac{\sum_{i=1}^n X_i}{n} \rightarrow \text{N}\Bigg(\mu, \dfrac{\sigma^2}{n}\Bigg)\]

Example of Corollary in use

Example 1

According to a large US study, the mean resting heart rate of adult women is about 74 beats per minutes (bpm), with standard deviation 13 bpm (NHANES 2003-2004).

1. Find the probability that the average resting heart rate for a random sample of 36 adult women is more than 3 bpm away from the mean.

2. Repeat the previous question for a single adult woman.

Example of CLT for exponential distribution

Example 2

Let \(X_i \sim Exp(\lambda)\) be iid RVs for \(i=1,2,\ldots,n\). Then \[\sum_{i=1}^n X_i \rightarrow\]

CLT for Discrete RVs

1. Binomial rv’s: Let \(X \sim Bin(n,p)\)

   • \(X = \displaystyle\sum_{i=1}^n X_i\), where \(X_i\) are iid \(\text{Bernoulli}(p)\)
   • Rule of thumb: \(np\geq10\) and \(n(1-p)\geq 10\) to use Normal approximation

2. Poisson rv’s: Let \(X \sim Poisson(\lambda)\)

   • \(X = \displaystyle\sum_{i=1}^n X_i\), where \(X_i\) are iid \(\text{Poiss}(1)\)
   • Recall from Chapter 18 that if \(X_i \sim Poiss(\lambda_i)\) and \(X_i\) independent, then \(\sum_{i=1}^n X_i \sim Poiss(\sum_{i=1}^n \lambda_i)\)
   • Rule of thumb: \(\lambda \geq10\) to use Normal approximation

At home example

Example 3

Suppose that the probability of developing a specific type of breast cancer in women aged 40-49 is 0.001. Assume the occurrences of cancer are independent. Suppose you have data from a random sample of 20,000 women aged 40-49.

1. How many of the 20,000 women would you expect to develop this type of breast cancer, and what is the standard deviation?

2. Find the exact probability that more than 15 of the 20,000 women will develop this type of breast cancer.

3. Use the CLT to find the approximate probability that more than 15 of the 20,000 women will develop this type of breast cancer.

4. Use the CLT to approximate the following probabilities, where \(X\) is the number of women that will develop this type of breast cancer.

   1. \(\mathbb{P}(15 \leq X \leq 22)\)

   2. \(\mathbb{P}(X > 20)\)

   3. \(\mathbb{P}(X < 20)\)

5. Find the approximate probability that more than 15 of the 20,000 women will develop this type of breast cancer - not using the CLT!

6. Use the CLT to approximate the approximate probability in the previous question!