chap9 - 9 Continuous Probability Distributions 9.1 General...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
9. Continuous Probability Distributions 9.1 General Terminology and Notation Continuous random variables have a range (set of possible values) an interval (or a collection of intervals) on the real number line. They have to be treated a little differently than discrete random variables because P ( X = x ) is zero for each x . Toi l lus t ra tearandomva r iab lew i tha continuous distribution , consider the simple spinning pointer in Figure 9.1. and suppose that all numbers in the 1 3 2 4 X Figure 9.1: Spinner: a device for generating a continuous random variable (in a zero-gravity, virtually frictionless environment) interval (0,4] are equally likely. The probability of the pointer stopping precisely at any given number x must be zero, because if each number has the same probability p> 0 , then the probability of R = { x :0 <x 4 } is the sum P (0 , 1] p = , since the set R is uncountably infinite. For a continuous random variable the probability of each point is 0 and probability functions cannot be used to describe a distribution. On the other hand, intervals of the same length h entirely contained in (0,4], for example the interval (0 , 1 4 ] and (1 3 4 , 1] all have the same probability (1/16 in this case). For 163
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
164 3 2 1 0 1 2 3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 * X Figure 9.2: continuous random variables we specify the probability of intervals, rather than individual points. Consider another example produced by choosing a “random point” in a region. Suppose we plot a graph a function f ( x ) as in Figure 9.2 (assume the function is positive and has finite integral) and then generate a point at random by closing our eyes and firing a dart from a distance until at least one lands in the shaded region under the graph. We assume such a point, here denoted "*" is “uniformly” distributed under the graph. This means that the point is equally likely to fall in any one of many possible regions of a given area located in the shaded region so we only need to know the area of a region to determine the probability that a point falls in it. Consider the x-coordinate X of the point "*" as our random variable (in Figure 9.2 it appears to be around 0 . 4) . Notice that the probability that X falls in a particular interval ( a, b ) is the measured by the area of the region above this interval, i.e. R b a f ( x ) dx and so the probability of any particular point P ( X = a ) is the area of the region immediately above this single point R a a f ( x ) dx =0 . This is another example of a random variable X which has a continuous distribution. For continuous X , there are two commonly used functions which describe its distribution. The first is the cumulative distribution function, used before for discrete distributions, and the second is the probability density function, the derivative of the c.d.f.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/01/2010 for the course MATH Stat 230 taught by Professor A during the Spring '07 term at Waterloo.

Page1 / 50

chap9 - 9 Continuous Probability Distributions 9.1 General...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online