Chapter 6Chapter 6

Continuous Random Continuous Random Variables and Variables and

Probability Probability DistributionsDistributions

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Continuous Random Continuous Random VariablesVariables

A random variable is continuous continuous if it can take any value in an interval.

Cumulative Distribution Cumulative Distribution FunctionFunction

The cumulative distribution functioncumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x, as a function of x

)()( xXPxF

Shaded Area is the Probability Shaded Area is the Probability That X is Between a and bThat X is Between a and b

x ba

Probability Density Function for Probability Density Function for a Uniform 0 to 1 Random a Uniform 0 to 1 Random

VariableVariable

0 1

1

x

f(x)

Areas Under Continuous Areas Under Continuous Probability Density FunctionsProbability Density Functions

Let X be a continuous random variable with the probability density function f(x) and cumulative distribution F(x). Then the following properties hold:

1. The total area under the curve f(x) = 1.2. The area under the curve f(x) to the left of

x0 is F(x0), where x0 is any value that the random variable can take.

Properties of the Probability Properties of the Probability Density FunctionDensity Function

0 1 xx0

f(x)

0

1Comments

Total area under the uniform probability density function is 1.

Properties of the Probability Properties of the Probability Density FunctionDensity Function

0 1 xx0

f(x)

0

1

Comments

Area under the uniform probability density function to the left of x0 is F(x0), which is equal to x0 for this uniform distribution because f(x)=1.

Reasons for Using the Reasons for Using the Normal DistributionNormal Distribution

1. The normal distribution closely approximates the probability distributions of a wide range of random variables.

2. Distributions of sample means approach a normal distribution given a “large” sample size.

3. Computations of probabilities are direct and elegant.

4. The normal probability distribution has led to good business decisions for a number of applications.

Probability Density Function Probability Density Function for a Normal Distributionfor a Normal Distribution

x0.0

0.1

0.2

0.3

0.4

Probability Density Function Probability Density Function of the Normal Distributionof the Normal Distribution

The probability density function for a probability density function for a normally distributed random variable Xnormally distributed random variable X is

Where and 2 are any number such that - < < and - < 2 < and where e and are physical constants, e = 2.71828. . . and = 3.14159. . .

xexf x -for 2

1)(

22 2/)(

2

Properties of the Normal Properties of the Normal DistributionDistribution

Suppose that the random variable X follows a normal distribution with parameters and 2. Then the following properties hold:

iii. The mean of the random variable is ,

iv. The variance of the random variable is 2,

v. The shape of the probability density function is a symmetric bell-shaped curve centered on the mean .

vi. By knowing the mean and variance we can define the normal distribution by using the notation

)(XE

22 ])[( XXE

),(~ 2NX

Effects of Effects of on the Probability on the Probability Density Function of a Normal Density Function of a Normal

Random VariableRandom Variable

x

0.0

0.1

0.2

0.3

0.4

1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Mean = 5 Mean = 6

Effects of Effects of 22 on the Probability on the Probability Density Function of a Normal Density Function of a Normal

Random VariableRandom Variable

x

0.0

0.1

0.2

0.3

0.4

1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Variance = 0.0625

Variance = 1

Cumulative Distribution Cumulative Distribution Function of the Normal Function of the Normal

DistributionDistribution

Suppose that X is a normal random variable with mean and variance 2 ; that is X~N(, 2). Then the cumulative distribution cumulative distribution functionfunction is

This is the area under the normal probability density function to the left of x0,. As for any proper density function, the total area under the curve is 1; that is F() = 1.

)()( 00 xXPxF

Shaded Area is the Probability Shaded Area is the Probability that X does not Exceed xthat X does not Exceed x00 for a for a

Normal Random VariableNormal Random Variable

xx0

f(x)

Range Probabilities for Range Probabilities for Normal Random VariablesNormal Random Variables

Let X be a normal random variable with cumulative distribution function F(x), and let a and b be two possible values of X, with a < b. Then

The probability is the area under the corresponding probability density function between a and b.

)()()( aFbFbXaP

Range Probabilities for Normal Range Probabilities for Normal Random VariablesRandom Variables

xb

f(x)

a

The Standard Normal The Standard Normal DistributionDistribution

Let Z be a normal random variable with mean 0 and variance 1; that is

We say that Z follows the standard normal distribution. Denote the cumulative distribution function as F(z), and a and b as two numbers with a < b, then

)1,0(~ NZ

)()()( aFbFbZaP

Standard Normal Distribution Standard Normal Distribution with Probability for z = 1.25with Probability for z = 1.25

z 1.25

0.8944

Finding Range Probabilities for Finding Range Probabilities for Normally Distributed Random Normally Distributed Random

VariablesVariables

Let X be a normally distributed random variable with mean and variance 2. Then the random variable Z = (X - )/ has a standard normal distribution: Z ~ N(0, 1)It follows that if a and b are any numbers with a < b, then

where Z is the standard normal random variable and F(z) denotes its cumulative distribution function.

aF

bF

bZ

aPbXaP )(

Computing Normal ProbabilitiesComputing Normal Probabilities

A very large group of students obtains test scores that are normally distributed with mean 60 and standard deviation 15. What proportion of the students obtained scores between 85 and 95?

0376.09525.09901.0

)67.1()33.2(

)33.267.1(

15

6095

15

6085)9585(

FF

ZP

ZPXP

That is, 3.76% of the students obtained scores in the range 85 to 95.