Binomial distribution to the normal distribution

Parametric equations

The binomial distribution $\mathrm{B}(n,p)$ is the probability of the number of successes $X$ in $n$ independent trials. When $n$ is large and $p$ is not extreme, $X$ can be approximated by the normal distribution with mean $np$ and variance $np(1-p)$. The standardized $Z$ is then approximately standard normal.

Normal approximation, after standardizing:

Density:

Interaction

• Trials n: hold Probability p fixed and drag $n$ from small to large. The bars move toward the bell curve
• X distribution: the continuous density with mean $np$ and variance $np(1-p)$ is drawn over the discrete bars
• Standardized Z: switch between $X$ and $Z$, and watch the mean and the scale settle
• Bernoulli trials: run one sequence of Bernoulli trials and compare the accumulated successes with the theoretical bars
• Sample: draw a new sequence
• Reset: clear that sequence

Related

• Conditional probability and Bayes’ theorem
• Pascal’s triangle
• Buffon’s needle

Further reading

• Binomial distribution (Wikipedia)
• De Moivre–Laplace theorem (Wikipedia)