Binomial Probability Distribution

Let #X# be a binomially distributed random variable with parameters #n# and #p#.
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To calculate the expected value of #X#, make use of the following formula:
\[\mu_{\small{X}} = \mathbb{E}[X]=n\cdot p\]
To calculate the variance of #X#, make use of the following formula:
\[\sigma^2_{\small{X}} = Var[X]=n\cdot p \cdot (1-p)\]
To calculate the standard deviation of #X#, make use of the following formula:
\[\sigma_{\small{X}} = SD[X]=\sqrt{n\cdot p \cdot (1-p)}\]

Sampling Distribution of the Sample Mean

If the conditions for normality are satisfied, the sampling distribution of the sample mean is approximately normally distributed with parameters:

• Expected value of the sample mean: #\mu_{\bar{X}} = \mu#
• Standard error of the sample mean: #\sigma_{\bar{X}} = \cfrac{\sigma}{\sqrt{n}}#

Sampling Distribution of the Sample Proportion

If the conditions for normality are satisfied, the sampling distribution of the sample proportion is approximately normally distributed with parameters:

• Expected value of the sample proportion: #\mu_{\hat{p}} = \pi#
• Standard error of the sample proportion: #\sigma_{\hat{p}} = \sqrt{\cfrac{\pi(1-\pi)}{n}}#