Random Variables zDiscrete Random Variables: a random variable that can assume only a countable number of values. The value of a discrete random variable comes from counting. z 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 zThe Continuous Uniform Distribution yA 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 zMean and standard deviation of the uniform probability density function:

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

Continuous Probability Distributions zThe Normal Distribution yA bell-shaped, continuous distribution with the following properties: xIt is unimodal; the normal distribution peaks at a single value. xIt is symmetrical; 50% of the area under the curve lies left of the center and 50% lies right of the center. xThe mean, mode, and median are equal. xIt 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:

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.