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IB Mathematics AI AHL A linear combination of n independent normal MAI Study Notes - New Syllabus

IB Mathematics AI AHL A linear combination of n independent normal MAI Study Notes

LEARNING OBJECTIVE

  • Central limit theorem.

Key Concepts: 

  • Central limit theorem.
  • A linear combination of n independent normal random variables

MAI HL and SL Notes – All topics

THE CENTRAL LIMIT THEOREM

The Central Limit Theorem (CLT)

Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with expected value $\mathbb{E}X_i = \mu < \infty$ and variance $0 < \text{Var}(X_i) = \sigma^2 < \infty$. Then, the random variable

$Z_n = \frac{\overline{X} – \mu}{\sigma/\sqrt{n}} = \frac{X_1 + X_2 + \ldots + X_n – n\mu}{\sqrt{n}\sigma}$

converges in distribution to the standard normal random variable as $n$ goes to infinity, that is

$\lim_{n \to \infty} P(Z_n \leq x) = \Phi(x), \qquad \text{for all } x \in \mathbb{R},$

where $\Phi(x)$ is the standard normal CDF.

Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$.

The sample mean $\overline{X}$ for $n$ independent samples has:

$E(\overline{X}) = \mu, \quad \text{Var}(\overline{X}) = \frac{\sigma^2}{n}, \quad \text{SD}(\overline{X}) = \frac{\sigma}{\sqrt{n}}$

Linear Combinations of Normal Random Variables

If $X \sim N(\mu, \sigma^2)$, then any linear combination, such as $\overline{X}$, is also Normally distributed:

$\overline{X} \sim N\left( \mu, \frac{\sigma^2}{n} \right)$

Example

Suppose $X \sim N(70, 9)$.

A sample of $n = 16$ gives $\overline{X} \sim N\left(70, \frac{9}{16} \right) = N(70, 0.5625)$.

Find $P(69 < \overline{X} < 71)$.

▶️Answer/Explanation

Solution:

Using a calculator:

$P(69 < \overline{X} < 71) = 0.954$

Statement (CLT)

For large sample sizes ($n > 30$), regardless of the original distribution of $X$, the distribution of the sample mean $\overline{X}$ approaches a Normal distribution:

$
\overline{X} \sim N\left( \mu, \frac{\sigma^2}{n} \right)
$

Example

Let $X \sim \text{Poisson}(10)$, $n = 40$. Then:

$
\mu = 10, \quad \sigma^2 = 10 \Rightarrow \overline{X} \sim N\left(10, \frac{10}{40}\right) = N(10, 0.25)
$

Find $P(9 < \overline{X} < 11)$:

▶️Answer/Explanation

Solution:

$P(9 < \overline{X} < 11) = 0.954$

The approximation improves as $n$ increases.

$n > 30$ is typically considered sufficient.

Example

Let $X \sim \text{Binomial}(100, 0.2)$, $n = 40$.

Then $\mu = 20$, $\sigma^2 = 16$, so:

$
\overline{X} \sim N(20, 0.4)
$

Find $P(\overline{X} < 19) = ?$

▶️Answer/Explanation

Solution:

$P(\overline{X} < 19) = 0.0569$

Using the Z-Table 

Convert to a standard normal variable:

$
Z = \frac{\overline{X} – \mu}{\sigma / \sqrt{n}}
$

Example

Population: $\mu = 100$, $\sigma = 30$, $n = 50$

$\overline{X} \sim N(100, 18)$

Find $P(90 < \overline{X} < 110)$:

▶️Answer/Explanation

Solution:

Convert to $Z$-scores and use Z-table or calculator:

$P = 0.982$

Applications 

  • Estimating population parameters
  • Hypothesis testing
  • Confidence intervals
  • Valid for skewed/non-normal distributions when $n$ is large

Example

You want to test whether a new teaching method increases test scores.

The current average is 70 with $\sigma = 12$.

A sample of 40 students has a mean of 74.

▶️Answer/Explanation

Solution:

$
Z = \frac{74 – 70}{12 / \sqrt{40}} \approx 2.11 \Rightarrow P \approx 0.017
$

Significant evidence that the method improves scores.

Visualizing Central Limit Theorem &  Practical Considerations

  • Histograms of sample means become bell-shaped as $n$ increases
  • Demonstrated using simulation
  • Independence: samples must be independent
  • Finite population correction if sampling without replacement
  • Not valid for very small samples from non-Normal populations

Example (conceptual):

Simulate 10,000 samples of size $n = 5$, $n = 30$, and $n = 100$ from a skewed distribution.
Plot histograms of sample means.
→ Distribution becomes approximately normal as $n$ increases.

Example:
For a small sample $n = 10$ from a skewed population, the CLT may not apply. Use non-parametric methods or increase $n$.

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