Chapter 8 Continuous Random Variables StatSlideShow Copyright 1994

Prof. Vinod Introduction 8 - 2

Continuum • The previous chapter was primarily devoted to random variables that were counts of some phenomenon that was called a “success. ” • These counts could only take on discrete values usually starting at zero. • In this chapter the focus will be on variables that take on any value in some continuum, where a continuum is simply a range of numbers on the real number line. 8 - 3

Example 1 • The range of newborn baby weights lies on a continuum between 2 and 13 pounds. [ ] 2 13 pounds • Observations measured in a continuum can be very close together; for example, two weights could be 8. 5672538764522134599. . . inches, and 8. 5672538764522134598. . . inches. 8 - 4

Discrete or Continuous? • Many continuous variables are discretized (rounded) by the instruments used to measure them. • It is important to distinguish the nature of the variable from its physical measurement. • Variables like heights and weights are continuous random variables even though they are usually given to the nearest unit. 8 - 5

Continuous vs. Discrete • One of the striking differences in discrete and continuous variables concerns the way in which probability is defined. • In a probability distribution for a discrete random variable, each possible outcome of the random variable is assigned its own probability. • However, for continuous random variables, there are infinitely many outcomes, and each has no probability. 8 - 6

Outcomes of a Continuous Random Variable • Outcomes of a continuous random variable do not have probability assigned to any one point, because there are too many points. • If we attempted to assign each value, even an infinitesimal probability, the sum of all the probabilities would exceed one. (sum >1? ) • Thus, for continuous random variables, probability is only assigned to intervals. 8 - 7

Probability is area, (strict=fuzzy) • Since probability is assigned to intervals when x is a continuous random variable (rv), the probability is associated with the area under the pdf along the specified interval. • Bottom line strict inequality and the fuzzy (e. g. ) are equivalent for continuous rv: • P(x<some constant) = P(x that same constant) 8 - 8

The Continuous Uniform Distribution 8 - 9

Continuous Uniform Distribution • Both uniform distributions distribute probability evenly across a sample space. • For the continuous uniform distribution, the probability density is spread out evenly over some range a to b as shown in the figure below. 8 - 10

Probability Density Functions (p. d. f) • Continuous random variables do not have probability distribution functions. • Instead, they have probability density functions which are denoted f(x). 8 - 11

The Uniform Probability Density Function The parameters of the density function are the minimum and maximum value of the random variable and are referred to as a and b, respectively. 8 - 12

Expected Value and Standard Deviation • The expected value (mean) of the continuous uniform random variable is • The standard deviation of the continuous uniform random variable is 8 - 13

Shape of the Continuous Uniform Distribution • When the uniform density function is graphed, it produces a rectangle or square. • The probability of observing a random variable in some interval is expressed as an area under the density function associated with the interval. 8 - 14

Calculating the Probability of an Interval Because the density function for the uniform distribution produces a rectangle, calculating the probability of an interval requires very simple geometry. Area of a rectangle =Height * width We want area to be 1, width from a to b is b-a, hence the height must be 1/(b-a) 8 - 15

Example 2 (Time to reach the fire) • The fire department records how long it takes each of its trucks to reach the scene of a fire. • Suppose that the distribution of arrival times is uniform with the minimum time being 2. 0 minutes and the maximum time being 15 minutes. • What is the probability that a truck will reach a reported fire scene within five to ten minutes? 8 - 16

Ex. 2 – Solution (5<time<10 minutes) Area = Height Width Height = Area = 8 - 17

Ex. 2 continued expected time. E(X), V(X) • What is the expected time until arrival? min. • What is the standard deviation of truck arrival times? min. 8 - 18

What does data from a uniform random variable look like? • While the density function for the uniform distribution has a flat top, a histogram of data from a uniformly distributed random process will not be so perfectly flat. 100 observations 1000 observations • When more observations are generated, the top of the distribution will begin to level. 8 - 19

The Normal Distribution 8 - 20

Normal Distribution • The normal distribution was originally called the Gaussian distribution, named after Karl Gauss who published a work in 1833 describing the mathematical definition of the distribution. • Gauss developed this distribution to describe the error in predicting the orbits of planets. 8 - 21

Shape of the Normal Distribution • Normal distributions are all bell shaped, but the bells come in various shapes and sizes. • However, since normal distributions are all bell shaped and symmetric, the mean, mode, and median are equal. mean median mode 8 - 22

Properties of the Normal Distribution Although the distribution can range in value from minus infinity to positive infinity, red values that are a great distance from the mean rarely occur. Range: -4<z<4 8 - 23

Properties of the Normal Distribution One of the more important properties of normal random variables is that within a fixed number of standard deviations from the mean, all Normal densities contain the same fraction of their probabilities. Remember one-sigma or twosigma rules? We now show they are related to area under the standard Normal density z~N(0, 1) curve. 8 - 24

Area Under the Curve • In the figure, the shaded area represents the probability of being within 1 s of the mean. • The probability of a normal random variable being in some interval corresponds to the shaded area under the curve. 8 - 25

Area Under the Curve • The total area under the curve from - to equals one. • The shaded area under the curve and the probability of being within one standard deviation ( 1 s) of the mean equals. 68. 8 - 26

Area Under the Curve 2 The shaded area in the figure above represents the probability of being within 2 s of the mean which equals. 9475 for every normal distribution. 8 - 27

Area Under the Curve 3 • The shaded area in the figure above represents the probability of being within 3 s of the mean which equals. 997 for every normal distribution. As you can see virtually all of the area under the curve is within three standard deviations of the mean. 8 - 28