Random Variables, Introduction to Binomial Distribution

If \(X\) is a discrete random variable with values \(x_1, x_2, x_3 \dots\) with probabilities \(p_1, p_2, p_3 \dots\), then:

Expected value (or mean)

$$\mu = E(x) =\sum x_ip_1$$

Variance

$$V(X) = \sigma^2 = \sum (x_i - \mu)^2 p_i$$

Standard Deviation

$$\sigma = \sqrt{\sum (x_i - \mu)^2 p_i}$$

The standard deviation has the same units as the original values.

The average distance of values from the mean.

Binomial Distribution for Random Variables

The binomial distribution can help you answer:

• What is the expected number of girls in six independent single births?
• What is the number of tall men in a random sample of 30 men from a large male population?

Required Information

• Number of trials (not a random value)
• Two possible outcomes
• Independent outcomes
• Probability of success \(p\) stays the same across events
• There is a probability of failure, which is \(1-p\).