CONTINUOUS RANDOM VARIABLES

Continuous random variables have values in a “continuum” of real numbers Examples -- X = How far you will hit a golf ball Y = How many hours you will spend studying Z = The weight of a watermelon W = Third quarter profits of a company

What is the probability you hit a golf ball exactly yards, yards?, 156 yards? For continuous random variables P(X = any specific #) = 0 POINT PROBABILITY

PROBABILITY DENSITY What is the probability you hit a golf ball between 100 and 200 yards, between 250 and 350 yards, between ….? Probability density f(x) is a measure (it is not a probability) of the likelihood you will get a value “around” x

PROPERTIES OF PROBABILITY DENSITY FUNCTIONS f(x) f(x)  0 for all values of x

INTERVAL PROBABILITIES P(a  X  b) = Area under the curve between a and b =

EXAMPLE If f(x) =.375x 2 for (0<x<2), find P(1  X  1.5) P(1  X  1.5)

DISCRETE E(X) =    xp(x) over all values of x CONTINUOUS Replace EXPECTED VALUE p(x) with f(x)dx

Mean -- Sample Calculation f(x) =.375x 2 for 0<X<2

VARIANCE

Example -- Variance f(x) =.375x 2 for 0<X<2

Approaches to Calculating Continuous Probabilities Probabilities are areas under density functions These areas are found using integral calculus Fortunately, the results for the most common continuous distributions can be found using: –Tables –Excel

REVIEW Continuous random variables are those that can assume any value in a continuous interval They are described by probability density functions Probabilities are areas under the density curve Means and variances are calculated the same way as that for discrete random variables except that p(x) is replaced by f(x)dx and the summation sign is replaced by the integral sign