Binomial Distribution, Introduction to Continuous Random Variables

Binomial Distribution

The probability of \(k\) successes in \(n\) independent binomial events with a probability of success of \(p\) is:

$$P(X=k) ={n \choose k}p^k (1 - p)^{n-k}$$

where

$${n \choose k} = \frac{n!}{k!(n-k)!}$$

This represents the number of ways to select \(k\) items from \(n\).

Mean

$$\mu = E(x) = np$$

Standard Deviation

The standard deviation of a binomial distribution is:

$$\sigma = \sqrt{np(1-p)}$$

Continuous Random Variable

A continuous random variable takes on all possible values in an interval. The probability corresponds to the area under a density curve, known as a probability density function.

Probability density functions must:

• Lie on or above the horizontal axis.
• The total area under the curve must be equal to 1.

This can be uniformly distributed if the graph is flat.