Normal Distributions, Normal Approximation to the Binomial

The expected value or \(E[x]\) is a "balancing point" or fulcrum for the distribution, if the density function had mass associated with it.

Normal Distribution

If the variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), it is denoted by:

$$X \sim N(\mu, \sigma)$$

If the standard deviation is smaller, then the distribution will be taller. It's more compact, but has to still maintain the same area of 1.0, so the height must increase to compensate.

The normal distribution is for continuous random variables. However, it is not the only distribution for continuous random variables.

Z-Score

A z-score is a way of transforming a value in a distribution to a standard normal random variable. This means that the variable now takes the form of:

$$z \sim N(0, 1)$$

The conversion is performed using the following formula:

$$z = \frac{x - \mu}{\sigma}$$

It's just a way to normalize values. Using technology (i.e., calculators) is a great way to find z-scores.

Percentiles

To find the value of a variable for which 90% of values fall below it, you need to:

• Use a z-table to find the z-score for which 90% of values fall under it.
• Convert the z-score to a value by using the following format:

$$x = z\sigma + \mu$$

Approximate Binomial Probabilities

Let's say that you want to find the number of people who are left handed in a sample of 120 people. This variable \(X\) has a binomial distribution with \(n=120, p = 0.10\). Therefore:

• \(\mu = np = 120(0.10) = 12\)
• \(\sigma = \sqrt{np(1-p)} = \sqrt{120(0.1)(0.9)} = 3.29\)

If \(n\) is large, which means that \(np \geq 10\) and \(n(1-p) \geq 10\), then you can model the distribution approximately with a normal distribution:

$$X \sim N\left(np, \sqrt{np(1-p)}\right)$$