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tags: introduction to probability, statistics
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([home](https://github.com/alexhkurz/introduction-to-probability/blob/master/README.md))

# Sampling Distribution of Sample Proportion

[sample-proportion.R](https://github.com/alexhkurz/introduction-to-probability/blob/master/src/sample-proportion.R) is essentially the same program as the one at [Khanacademy - Sampling distribution of sample proportion part 2](https://www.khanacademy.org/math/ap-statistics/sampling-distribution-ap/sampling-distribution-proportion/v/sampling-distribution-of-sample-proportion-part-2?modal=1), see there for explanations.

Modify the program to run your own experiments.

**Remark:** The distribution of sample proportions is $X_n\sim B(n,p)/n$, where $B(n,p)$ is the binomial distribution. [^bernoulli]

[^bernoulli]: For example, $p$ is the probability of drawing a green ball in an urn and $n$ is the sample size. Then the limit of $X_n$ for $n\to\infty$ is normal. This does not directly follow from the central limit theorem as we discussed it, since $X_n\sim B(n,p)/n$ is not taking a mean over *identical* independent distributions.