<p>If <span class="math-container">$X_1,X_2,\ldots, X_n$</span> is an i.i.d sample drawn from Bernoulli<span class="math-container">$(p)$</span> distribution, where <span class="math-container">$p$</span> is the population parameter, then the distribution of <span class="math-container">$\displaystyle\sum_{i=1}^{n}X_i$</span> is <span class="math-container">$\text{Binomial}(n,p)$</span>, where <span class="math-container">$n$</span> is the sample size, and <span class="math-container">$p$</span> is the population parameter. For <span class="math-container">$n$</span> large enough,
by CLT, the approximate distribution of the sample sum <span class="math-container">$\displaystyle\sum_{i=1}^{n}X_i$</span> is <span class="math-container">$\text{Normal}$</span> with mean <span class="math-container">$np$</span> and variance <span class="math-container">$np(1-p)$</span> and the approximate distribution of the sample proportion <span class="math-container">$\displaystyle\dfrac{\sum_{i=1}^{n}X_i}{n}$</span> is <span class="math-container">$\text{Normal}$</span> with mean <span class="math-container">$p$</span> and variance <span class="math-container">$\dfrac{p(1-p)}{n}$</span>.</p>

CC BY-SA 4.0
2024-09-19T13:07:13.757Z
2024-09-19T13:07:13.757Z
Amit
<p>The sample proportion follows an approximately normal distribution provided certain conditions are met.</p>
<p>In other cases, the distribution is binomial (Source: <a href="https://ecampusontario.pressbooks.pub/introstats/chapter/6-3-sampling-distribution-of-the-sample-proportion/" rel="nofollow noreferrer">https://ecampusontario.pressbooks.pub/introstats/chapter/6-3-sampling-distribution-of-the-sample-proportion/</a>).</p>
<p>Question: A binomial distribution has 2 parameters. What are the parameters (number of trials and probability of success) of the binomial variable (sample proportion) here? I understand that the probability of success is the population proportion. Just can't figure out the other parameter.</p>

CC BY-SA 4.0
2024-09-19T12:47:02.030Z
I’m voting to close this question because it is not about economics and belongs on Cross Validated instead.
Richard Hardy
<probability>
Parameters of the sampling distribution of the sample proportion
AcadEconInd
# Parameters of the sampling distribution of the sample proportion

(Asked by: username_0 on 2024-09-19)

The sample proportion follows an approximately normal distribution provided certain conditions are met.

In other cases, the distribution is binomial (Source: <https://ecampusontario.pressbooks.pub/introstats/chapter/6-3-sampling-distribution-of-the-sample-proportion/>).

Question: A binomial distribution has 2 parameters. What are the parameters (number of trials and probability of success) of the binomial variable (sample proportion) here? I understand that the probability of success is the population proportion. Just can't figure out the other parameter.

**Tags:** probability

> **Comments:**
> **username_1**: I’m voting to close this question because it is not about economics and belongs on Cross Validated instead.  

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## 1 answers

### Answer by username_2 on 2024-09-19. Score: 1

If $X\_1,X\_2,\ldots, X\_n$ is an i.i.d sample drawn from Bernoulli$(p)$ distribution, where $p$ is the population parameter, then the distribution of $\displaystyle\sum\_{i=1}^{n}X\_i$ is $\text{Binomial}(n,p)$, where $n$ is the sample size, and $p$ is the population parameter. For $n$ large enough,
by CLT, the approximate distribution of the sample sum $\displaystyle\sum\_{i=1}^{n}X\_i$ is $\text{Normal}$ with mean $np$ and variance $np(1-p)$ and the approximate distribution of the sample proportion $\displaystyle\dfrac{\sum\_{i=1}^{n}X\_i}{n}$ is $\text{Normal}$ with mean $p$ and variance $\dfrac{p(1-p)}{n}$.


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