Study Modules (PPT presentations):

Introduction to Continuous Probability Distributions

Normal Probability Distribution

Discrete Distributions

Excel Tutorial:

Computing Normal Probabilities

Java Applet:

Normal Distribution Areas

Normal Approximation to Binomial Probabilities

Continuous Random Variables:

A continuous random variable can assume ____any value_______________ in an interval on the real line or in a collection of intervals.

It is not possible to talk about the probability of the random variable assuming a __specific___________ (P(x=X)=0) value.

Instead, we talk about the probability of the random variable assuming a value within a given _____interval_____________________.

The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the ___area under the graph_______ of the ___probability density function__ between x1 and x2.

Three continuous Random Variables will be discussed in this chapter: Uniform Probability Distribution, Normal Probability Distribution, and Exponential Probability Distribution.

I. Uniform Probability Distribution

A random variable is uniformly distributed whenever the ____probability__________ is proportional to the interval’s length.

The uniform probability ___________________________ is:

where: a = smallest value the variable can assume b = largest value the variable can assume

Expected Value of x:

E(x) = ______________________________

Variance of x:

Var(x) = ______________________________

Example: Slater’s Buffet:

Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces.

Therefore:

The amount salad customer taken is an uniform distributed random variable with density function: f(x) = 1/10, for 5 ≤ x ≤ 15 = 0, elsewhere Where: x= salad plate filling weight Expected Value of x: E(x) = (a+b)/2 = __µ (mean)_________________ Variance of x: Var(x) = (b-a)2/12 = __ σ_(standard deviation)______________

Area as a Measure of Probability:

The area under the graph of f(x) and probability are identical.

This is valid for all continuous random variables.

The probability that x takes on a value between some lower value x1 and some higher value x2 can be found by computing the area under the graph of f(x) over the interval from x1 to x2.

II. Normal Probability Distribution

The normal probability distribution is the most important distribution for describing a continuous random variable.

It is widely used in statistical inference.

It has been used in a wide variety of applications including:

Heights of people

Test scores

Rainfall amounts

Scientific measurements

Abraham de Moivre, a French mathematician, published The Doctrine of Chances in 1733. He derived the normal distribution.

Standard Normal Probability Distribution:

A random variable having a normal distribution with a mean of _____ and a standard deviation of ____ is said to have a standard normal probability distribution.

Converting to the Standard Normal Distribution:

z = ____ (X - μ) / σ ______________________________________

We can think of z as a measure of the number of standard deviations x is from µ.

Reference: Using Excel to Computer Standard Normal Probabilities (p.266)

Example: Pep Zone

Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of