Random Variables

Discrete Random Variables:a random variable that can assume only a countable number of values. The value of a discrete random variable comes from counting.

Continuous Random Variable:random variables that can assume any value on a continuum. Measurement is required to determine the value for a continuous random variable.

Continuous Probability Distributions

The probability distribution of a continuous random variable is represented by a probability density function that defines a curve. The area under the curve corresponds to the probabilities for the random variable.

Continuous Probability Distributions

The Continuous Uniform Distribution A probability distribution in which the

probability of a value occurring between two points, a and b, is the same as the probability between any other two points, c and d, given that the distance between a and b is equal to the distance between c and d.

f (x) = 1 / ( b - a ) if a < x < b

The Uniform Probability Density Function

Mean and standard deviation of the uniform probability density function:

a b

b a2

12

EXAMPLE

Suppose the research department of a steel manufacturer believes that one of the company’s rolling machines is producing sheets of steel of varying thickness. The thickness is a uniform random variable with values between 150 and 200 millimeters. Any sheet less than 160 millimeters must be scrapped because they are unacceptable to buyers.

EXAMPLE

Calculate the mean and standard deviation of x, the thickness of the sheets produced by this machine. Then graph the probability distribution and show the mean on the horizontal axis.

Calculate the fraction of steel sheets produced by this machine that have to be scrapped.

Continuous Probability Distributions

The Normal Distribution A bell-shaped, continuous distribution with

the following properties:It is unimodal; the normal distribution peaks at a

single value.It is symmetrical; 50% of the area under the

curve lies left of the center and 50% lies right of the center.

The mean, mode, and median are equal.It is asymptotic; the normal distribution

approaches the horizontal axis on each side of the mean toward +

The Normal Distribution

The Normal Distribution is defined by two parameters:

X

X

M ean

Variance

2

The Standard Normal Distribution

The Standard Normal Distribution is a continuous, symmetrical, bell-shaped distribution that has a mean of 0 and a standard deviation of 1.

The Z Score

The Z Score is the number of standard deviations between the mean and the point X.

ZX X

X