Statistikk og Simulering

# Økt 13. Estimation of the Mean

## The Central Limit Theorem

#### 14.4. The Central Limit Theorem

Let $X$ be a stochastic variable, uniformly distributed on the

Øving 14.7 Make a Matlab routine sample(m) which takes an integer argument $m$, draws $m$ random numbers uniformly distributed on the integer range $\left[0,10\right]$, and calculates the sample mean of these $m$ numbers.

Øving 14.8 Run sample(1) 100 times and draw a histogram of the values. Repeat this for sample(m) for $m=5,25,125,500$.

Compare the histograms for different values of $m$. What do you see?

Let $X={X}_{1}+{X}_{2}+\dots +{X}_{n}$ be a sum of identically distributed variables ${X}_{i}$. regardless of the exact distribution of ${X}_{i}$

When as $n\to \infty$, $X$ has always the same distribution, namely the normal distribution.

Tabell 2: The Central Limit Theorem

Øving 14.9 Consider the Central Limit Theorem (CLT) in Table 2. What evidence do you see of the CLT in the histograms in the previous exercise.

It is the CLT which allows us to use the normal distribution to approximate the binomial distribution for large $n$. It also means that when the sample size is large, we can make confidence intervals for the mean based on the normal distribution, regardless of the distribution of the underlying variables.