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Binomial Distribution

When modeling a situation where there are n independent trials with a constant probability p of "success" in each test we use a binomial distribution.

For example, if we toss n equal coins and we count heads as success, the probability of getting head can be any value between 0 and 1.

A binomial distribution is characterized by two parameters: n (a natural number) and p a number between 0 and 1.

If a random variable X follows the binomial distribution with parameters n and p, we write

The probability of getting exactly k successes in n trials is given by the probability mass function (or probability density function):


The later expression is known as the binomial coefficient, "n choose k", or the number of possible ways to choose k successes from n observations. The binomial coefficients form the rows of Pascal's triangle and can be calculated using factorials:

When p = 0.5 the probability mass function is symmetric:

Distribución Binomial: función de densidad o masa simétrica | matematicasVisuales

In other cases the function is asymmetric:

Distribución Binomial: función de densidad o masa asimétrica| matematicasVisuales

Mean and Variance of the Binomial Distribution are:

This applet is developed in HTML5 and can be managed using the mouse or the touchscreen of your tablet or mobile phone.

We can change the parameter n of the binomial distribution. The mean is represented by a triangle and it can be seen as a point of equilibrium. By dragging it we modify the mean and the parameter p.

Red points control vertical and horizontal scales.

We can show a normal curve that has the same mean and variance as the binomial distribution.

In some cases, this normal curve is close to the binomial and can be used for calculations.

Distribución Binomial: curva normal como buena aproximación | matematicasVisuales

You can see why it is recommended to extend the interval for which you want to calculate the probability of the binomial in 0.5 above and below to use the normal distribution to approximate the probability (correction for continuity adjustement).

For example:

Binomial distribution, normal approximation example formula| matematicasVisuales
Distribución Binomial: curva normal corrección de continuidad | matematicasVisuales

In other cases, this normal curve is not a good approximation to the binomial distribution:

Binomial distribution: normal curve is not a good approximation | matematicasVisuales

Even with the correction for continuity adjustement the approximation is not accurate:

Binomial distribution: normal curve not a good approximation even with correction for continuity adjustement| matematicasVisuales

You can see more about the Normal approximation to the Binomial.


Normal distributions
The Normal distribution was studied by Gauss. This is a continuous random variable (the variable can take any real value). The density function is shaped like a bell.
Normal Distributions: One, two and three standard deviations
One important property of normal distributions is that if we consider intervals centered on the mean and a certain extent proportional to the standard deviation, the probability of these intervals is constant regardless of the mean and standard deviation of the normal distribution considered.
Normal Distributions: (Cumulative) Distribution Function
The (cumulative) distribution function of a random variable X, evaluated at x, is the probability that X will take a value less than or equal to x. In this page we study the Normal Distribution.
Normal Distributions: Probability of Symmetric Intervals
Calculating probabilities of symmetric intervals around the mean of a normal distribution.
Poisson distribution
Poisson distribution is discrete (like the binomial) because the values that can take the random variable are natural numbers, although in the Poisson distribution all the possible cases are theoretically infinite.
Student's t-distributions
Student's t-distributions were studied by William Gosset(1876-1937) when working with small samples.