Sampling Distributions

Statistics

• Statistics

• Sampling Distributions

• Central Limit Theorem

Sample

When collecting data to construct a sample, the sample is a collection of random variables.

Therefore, the sample can be subjected to probability properties.

iid Random Variables

A sample of random variables are said to be iid if they are identical and independentally distributed.

For example, \(X\) and \(Y\) are iid, if \(X\) and \(Y\) has the same distribution \(f(\theta)\) and \(X \perp Y\)

Statistics

A statistic is a transformation of the the sample data.

Before data is calculated, a statistic from a sample can take any value.

Therefore, a statistic must be a random variable.

Sampling Distributions

• Statistics

• Sampling Distributions

• Central Limit Theorem

Sampling Distributions

A sampling distribution is the distribution of a statistic. Many known statistics have a known distribution.

\(\bar X\)

Let \(X_1, X_2, \ldots, X_n\overset{iid}{\sim}N(\mu,\sigma^2)\) , show that \(\bar X \sim N(\mu,\sigma^2/n)\). Note: the MGF of \(X_i\) is \(e^{\mu t + \frac{t^2\sigma^2}{2}}\).

Sum of \(\chi^2_1\)

Let \(Z_1^2, \ldots, Z_n^2\) be a iid \(\chi^2_1\). Find \(Y = \sum^n_{i=1} Z_i^2\)

\(s^2\)

t-distribution

Let \(Z\sim N(0,1)\), \(W\sim \chi^2_\nu\), \(Z\perp W\); therefore:

\[ T=\frac{Z}{\sqrt{W/\nu}} \sim t_\nu \]

F-distribution

Let \(W_1\sim\chi^2_{\nu_1}\) \(W_2\sim\chi^2_{\nu_2}\), and \(W_1\perp W_2\); therefore:

\[ F = \frac{W_1/\nu_1}{W_2/\nu_2}\sim F_{\nu_1,\nu_2} \]

Central Limit Theorem

• Statistics

• Sampling Distributions

• Central Limit Theorem

Central Limit Theorem

Let \(X_1, X_2, \ldots, X_n\) be identical and independent distributed random variables with \(E(X_i)=\mu\) and \(Var(X_i) = \sigma²\). We define

\[ Y_n = \sqrt n \left(\frac{\bar X-\mu}{\sigma}\right) \mathrm{ where }\ \bar X = x\frac{1}{n}\sum^n_{i=1}X_i. \]

Then, the distribution of the function \(Y_n\) converges to a standard normal distribution function as \(n\rightarrow \infty\).

Central Limit Theorem

\[ \bar X \sim N\left(\mu, \frac{\sigma^2}{n}\right) \]

Example

Let \(X_1, \ldots, X_n \overset{iid}{\sim} \chi^2_p\), the MGF is \(M(t)=(1-2t)^{-p/2}\). Find the distribution of \(\bar X\) as \(n \rightarrow \infty\).