Chapter 05

Chapter 5: Continuous Random Variables

McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

2

Where We’ve Been Using probability rules to find the probability of discrete events

Examined probability models for discrete random variables


3

Where We’re Going Develop the notion of a probability distribution for a continuous random variable

Examine several important continuous random variables and their probability models

Introduce the normal probability distribution

5.1: Continuous Probability Distributions A continuous random variable can assume any numerical value within some interval or intervals.

The graph of the probability distribution is a smooth curve called a probability density function, frequency function or probability distribution.

4McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

5.1: Continuous Probability Distributions There are an

infinite number of possible outcomes p(x) = 0 Instead, find

p(a<x<b) Table Software Integral

calculus)


5.2: The Uniform Distribution

X can take on any value between c and d with equal probability= 1/(d - c)

For two values a and b

dbaccdabbxaP

)(



Mean:

Standard Deviation: 12

2

cd

dc



Suppose a random variable x is distributed uniformly with

c = 5 and d = 25.What is P(10 x 18)?


8


Suppose a random variable x is distributed uniformly with

c = 5 and d = 25.What is P(10 x 18)?


9

40.5251018)1810(

xP

The probability density function f(x):

µ = the mean of x = the standard deviation of x = 3.1416…e = 2.71828 …

5.3: The Normal Distribution

Closely approximates many situations Perfectly symmetrical around its mean

2]/)[( 2

21)(

x

exf



22

21)(

z

ezf

Each combination of µ and produces a unique normal curve

The standard normal curve is used in practice, based on the standard normal random variable z (µ = 0, = 1), with the probability distribution

The probabilities for z are given in Table IV11McClave: Statistics, 11th ed. Chapter 5:

Continuous Random Variables


0531.3413.3944.)00.10()25.10(

)25.11(

6826.3413.3413.)11(

3413.)000.1(

3413.)00.10(

zPzPzP

zP

zP

zP



For a normally distributed random variable x, if we know µ and ,

ii

xz


So any normally distributed variable can be analyzed with this single distribution


Say a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3,000 and = 50)

What is the probability that the car will go more than 3,100 yards without recharging?



0228.4772.5.)00.20(5.

)00.2(5.0)00.2(5030003100)3100(

zPzPzP

zPxP

Say a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3,000 and = 50)

What is the probability that the car will go more than 3,100 yards without recharging?



To find the probability for a normal random variable … Sketch the normal distribution Indicate x’s mean Convert the x variables into z values Put both sets of values on the sketch, z

below x Use Table IV to find the desired

probabilities


5.4: Descriptive Methods for Assessing Normality

If the data are normal A histogram or stem-and-leaf display will look

like the normal curve The mean ± s, 2s and 3s will approximate the

empirical rule percentages The ratio of the interquartile range to the

standard deviation will be about 1.3 A normal probability plot , a scatterplot with the

ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis, will produce close to a straight line


5.4: Descriptive Methods for Assessing NormalityErrors per MLB team in 2003

Mean: 106 Standard Deviation: 17

IQR: 22

29.11722

s

IQR

15755511063

14072341062

1238917106

sx

sx

sx22 out of 30: 73%

28 out of 30: 93%

30 out of 30: 100%


5.4: Descriptive Methods for Assessing Normality

A normal probability plot is a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis


5.5: Approximating a Binomial Distribution with the Normal Distribution

Discrete calculations may become very cumbersome

The normal distribution may be used to approximate discrete distributions The larger n is, and the closer p is

to .5, the better the approximation Since we need a range, not a value,

the correction for continuity must be used A number r becomes r+.5


Calculate the mean plus/minus 3 standard deviations

npqnp 3 If this interval is in the range 0 to n, the approximation will be reasonably closeExpress the binomial probability as a range of values

)()()(

axPbxPaxP

Find the z-values for each binomial value

)5.(az Use the standard normal distribution to find

the probability for the range of values you calculated21McClave: Statistics, 11th ed. Chapter 5:

Continuous Random Variables


Flip a coin 100 times and compare the binomial and normal results

0796.)10.010.0(5505.50

5505.49)5.505.49(

55.5.100505.100

0796.5.5.50100)50( 5050

zP

zPxP

xP


Binomi

al:Nor

mal:


Flip a weighted coin [P(H)=.4] 10 times and compare the results

1255.)32.032.0(55.145.5

55.145.4)5.55.4(

55.16.4.1044.10

1204.6.4.510)5( 55

zP

zPxP

xP


Binomi

al:Nor

mal:


Flip a weighted coin [P(H)=.4] 10 times and compare the results

1255.)32.032.0(55.145.5

55.145.4)5.55.4(

55.16.4.1044.10

1204.6.4.510)5( 55

zP

zPxP

xP


Binomi

al:Nor

mal:


The more p differs from .5, and the smaller n is,the less precise the approximation will be

5.6: The Exponential Distribution


Probability Distribution for an Exponential Random Variable x Probability Density Function

Mean: µ =

Standard Deviation: =

)0(1)( / xexf x

5.6: The Exponential DistributionSuppose the waiting time to see the nurse at the student health center is distributed exponentially with a mean of 45 minutes. What is the probability that a student will wait more than an hour to get his or her generic pill?


2660

5.6: The Exponential DistributionSuppose the waiting time to see the nurse at the student health center is distributed exponentially with a mean of 45 minutes. What is the probability that a student will wait more than an hour to get his or her generic pill?


27

2645.)60(

)(

33.1

4560

eexP

eaxPa

60

Documents

Chapter 05