Chapter 5 Terms

Standard Normal Random Variation

A random variable with a standard normal distribution, denoted by the symbol z

Parameters of normal distribution

* Effects of varying parameters * Mean, µ (mu), and standard deviation, δ (delta)

Standard Normal Distribution

* Normal distribution with µ=0 and δ=1 * A random variable with a standard normal distribution, denoted by the symbol z, is called a standard normal random variable.

Bell-shaped probability distribution

*Also known as a bell curve; one of the most commonly observed continuous random variables has this. Known as a normal random variable *Mean, median, and mode are equal; the middle line is equal to this

Property of Normal Distribution

*If x is a normal random variable with mean μ and standard deviation δ, then the random variable z, defined by the formula z=(x-µ)/δ has a standard normal distribution. *The value z describes the number of standard deviations between x and µ

Steps for finding a Probability corresponding to a Normal Random Variable X

1. Sketch the normal curve (bell curve) and indicate the mean of the random variable X. 2. Shade the area corresponding to the probability of interest. 3. Convert the boundaries (x values) of the shaded area to z values by using z = (x-µ) / δ 4. Use the Z - table to find the areas corresponding to the z values.

Steps for finding a x-value correspond to a specific probability

1. Sketch the normal curve associated with the variable. 2. Shade the specific area of interest. 3. Use Z-table to find Z0 corresponding to the probability. 4. Convert z value to x-value by using: z = (x-µ) / δ (or x = µ+zδ)

Properties of standard normal curve

1. The formula for the probability distribution of z is (top picture) 2. The standard normal distribution is symmetric about its mean 0 3. The total area under the standard normal probability distribution equals 1 4. The area under normal curve is equal to the associated area under the standard normal curve (bottom picture)

Smooth Curve

The graphical form of the probability distribution for a continuous random variable x

Continuous Random Variables

can assume any value contained in one or more intervals

Probability Density Function

PDF *Also known as a frequency function; the curve of the probability distribution for a continuous random variable *The probability that x falls between two values, a and b, (i.e., P(a < x < b)), is the area under the curve between a and b *The total area under a probability distribution curve equals 1 *Since an area under the curve requires an interval to have any area at all, we cannot find the probability of a single point for a continuous distribution. *p(a)=p(b)=0 *p(a<x<b)=p(a≤x≤b)

Normal Probability Distribution

The probability of a bell curve

Probability distribution for a continuous random variable, x

can be represented by a smooth curve - a function of x, denoted by f(x)