Chapter 6Chapter 6

Continuous Random Continuous Random Variables and Probability Variables and Probability

DistributionsDistributions

©

Continuous Random Continuous Random VariablesVariables

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

Cumulative Distribution Cumulative Distribution FunctionFunction

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

Cumulative Distribution Cumulative Distribution FunctionFunction

0 1

1

F(x)

Cumulative Distribution Function for a Random variable Over 0 to 1

Cumulative Distribution Cumulative Distribution FunctionFunction

Let X be a continuous random variable Let X be a continuous random variable with a cumulative distribution function with a cumulative distribution function F(x), and let a and b be two possible F(x), and let a and b be two possible values of X, with a < b. The values of X, with a < b. The probability probability that X lies between a and bthat X lies between a and b is is

)()()( aFbFbXaP

Probability Density FunctionProbability Density Function

Let X be a continuous random variable, and let x be any Let X be a continuous random variable, and let x be any number lying in the range of values this random variable can number lying in the range of values this random variable can take. The take. The probability density functionprobability density function, f(x), of the random , f(x), of the random variable is a function with the following properties:variable is a function with the following properties:

1.1. f(x) > 0 for all values of xf(x) > 0 for all values of x2.2. The area under the probability density function The area under the probability density function f(x)f(x) over all over all

values of the random variable X is equal to 1.0values of the random variable X is equal to 1.03.3. Suppose this density function is graphed. Let a and b be two Suppose this density function is graphed. Let a and b be two

possible values of the random variable X, with a<b. Then the possible values of the random variable X, with a<b. Then the probability that X lies between a and b is the area under the probability that X lies between a and b is the area under the density function between these points.density function between these points.

4.4. The cumulative density function F(xThe cumulative density function F(x00) is the area under the ) is the area under the probability density function probability density function f(x)f(x) up to x up to x00

where xwhere xm m is the minimum value of the random variable x.is the minimum value of the random variable x.

0

)()( 0

x

xm

dxxfxf

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

0 1

1

x

f(x)

Areas Under Continuous Probability Areas Under Continuous Probability Density FunctionsDensity Functions

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

1.1. The total area under the curve f(x) = 1.The total area under the curve f(x) = 1.

2.2. The area under the curve f(x) to the left The area under the curve f(x) to the left of xof x00 is F(x is F(x00), where x), where x00 is any value that is any value that the random variable can take.the random variable can take.

Properties of the Probability Density Properties of the Probability Density FunctionFunction

0 1 xx0

f(x)

0

1Comments

Total area under the uniform probability density function is 1.

Properties of the Probability Density Properties of the Probability Density FunctionFunction

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.

Rationale for Expectations of Rationale for Expectations of Continuous Random VariablesContinuous Random Variables

Suppose that a random experiment leads Suppose that a random experiment leads to an outcome that can be represented by to an outcome that can be represented by a continuous random variable. If N a continuous random variable. If N independent replications of this independent replications of this experiment are carried out, then the experiment are carried out, then the expected valueexpected value of the random variable is of the random variable is the average of the values taken, as the the average of the values taken, as the number of replications becomes infinitely number of replications becomes infinitely large. The expected value of a random large. The expected value of a random variable is denoted by variable is denoted by E(X).E(X).

Rationale for Expectations of Rationale for Expectations of Continuous Random VariablesContinuous Random Variables

Similarly, if g(x) is any function of the Similarly, if g(x) is any function of the random variable, X, then the expected value random variable, X, then the expected value of this function is the average value taken of this function is the average value taken by the function over repeated independent by the function over repeated independent trials, as the number of trials becomes trials, as the number of trials becomes infinitely large. This expectation is denoted infinitely large. This expectation is denoted E[g(X)]. By using calculus we can define E[g(X)]. By using calculus we can define expected values for continuous random expected values for continuous random variables similarly to that used for discrete variables similarly to that used for discrete random variables.random variables.

x

dxxfxgxgE )()()]([

Mean, Variance, and Standard Mean, Variance, and Standard DeviationDeviation

Let X be a continuous random variable. There are two important Let X be a continuous random variable. There are two important expected values that are used routinely to define continuous expected values that are used routinely to define continuous probability distributions.probability distributions.

i.i. The The mean of Xmean of X, denoted by , denoted by XX, is defined as the expected value of , is defined as the expected value of X.X.

ii.ii. The The variance of Xvariance of X, denoted by , denoted by XX22, is defined as the expectation , is defined as the expectation

of the squared deviation, (X - of the squared deviation, (X - XX))22, of a random variable from its , of a random variable from its meanmean

Or an alternative expression can be derivedOr an alternative expression can be derived

iii.iii. The The standard deviation of Xstandard deviation of X, , XX, is the square root of the , is the square root of the variance.variance.

)(XEX

])[( 22XX XE

222 )( XX XE

Linear Functions of VariablesLinear Functions of VariablesLet X be a continuous random variable with Let X be a continuous random variable with mean mean XX and variance and variance XX

22, and let a and b any , and let a and b any constant fixed numbers. Define the random constant fixed numbers. Define the random variable W asvariable W as

Then the mean and variance of W areThen the mean and variance of W are

andand

and the standard deviation of W isand the standard deviation of W is

bXaW

XW babXaE )(

222 )( XW bbXaVar

XW b

Linear Functions of VariableLinear Functions of Variable

An important special case of the previous An important special case of the previous results is the standardized random results is the standardized random variablevariable

which has a mean 0 and variance 1.which has a mean 0 and variance 1.X

XXZ

Reasons for Using the Normal Reasons for Using the Normal DistributionDistribution

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

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

3.3. Computations of probabilities are direct Computations of probabilities are direct and elegant.and elegant.

4.4. The normal probability distribution has led The normal probability distribution has led to good business decisions for a number of to good business decisions for a number of applications.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 The probability density function for a probability density function for a normally distributed random variable Xnormally distributed random variable X is is

Where Where and and 22 are any number such that are any number such that -- < < < < and - and - < < 22 < < and where e and where e and and are physical constants, e = are physical constants, e = 2.71828. . . and 2.71828. . . and = 3.14159. . . = 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 Suppose that the random variable X follows a normal distribution with parameters distribution with parameters and and 22. Then the . Then the following properties hold:following properties hold:

i.i. The mean of the random variable is The mean of the random variable is ,,

ii.ii. The variance of the random variable is The variance of the random variable is 22,,

iii.iii. The shape of the probability density function is a The shape of the probability density function is a symmetric bell-shaped curve centered on the mean symmetric bell-shaped curve centered on the mean as shown in Figure 6.8.as shown in Figure 6.8.

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

)(XE

22 ])[( XXE

),(~ 2NX

Effects of Effects of on the Probability Density on the Probability Density Function of a Normal Random VariableFunction of a Normal Random 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 Density on the Probability Density Function of a Normal Random VariableFunction of a Normal Random 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 Function Cumulative Distribution Function of the Normal Distributionof the Normal Distribution

Suppose that X is a normal random Suppose that X is a normal random variable with mean variable with mean and variance and variance 22 ; ; that is X~N(that is X~N(, , 22). Then the ). Then the cumulative cumulative distribution functiondistribution function is is

This is the area under the normal This is the area under the normal probability density function to the left of probability density function to the left of xx00, as illustrated in Figure 6.10. As for , as illustrated in Figure 6.10. As for any proper density function, the total any proper density function, the total area under the curve is 1; that is F(area under the curve is 1; that is F() = 1.) = 1.

)()( 00 xXPxF

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

Random VariableRandom Variable

xx0

f(x)

Range Probabilities for Range Probabilities for Normal Random VariablesNormal Random Variables

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

The probability is the area under the The probability is the area under the corresponding probability density function corresponding probability density function between a and b.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 Let Z be a normal random variable with mean 0 and variance 1; that ismean 0 and variance 1; that is

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

)1,0(~ NZ

)()()( aFbFbZaP

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

z 1.25

0.8944

Finding Range Probabilities for Normally Finding Range Probabilities for Normally Distributed Random VariablesDistributed Random Variables

Let X be a normally distributed random variable with Let X be a normally distributed random variable with mean mean and variance and variance 22. Then the random variable Z = . Then the random variable Z = (X - (X - )/)/ has a standard normal distribution: Z ~ N(0, has a standard normal distribution: Z ~ N(0, 1)1)It follows that if a and b are any numbers with a < b, It follows that if a and b are any numbers with a < b, thenthen

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

aF

bF

bZ

aPbXaP )(

Computing Normal ProbabilitiesComputing Normal Probabilities

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

Approximating Binomial Probabilities Approximating Binomial Probabilities Using the Normal DistributionUsing the Normal Distribution

Let X be the number of successes from n Let X be the number of successes from n independent Bernoulli trials, each with probability independent Bernoulli trials, each with probability of success of success . The number of successes, X, is a . The number of successes, X, is a Binomial random variable and if nBinomial random variable and if n(1 - (1 - ) > 9 a ) > 9 a good approximation isgood approximation is

Or if 5 < nOr if 5 < n(1 - (1 - ) < 9 we can use the continuity ) < 9 we can use the continuity correction factor to obtaincorrection factor to obtain

where Z is a standard normal variable.where Z is a standard normal variable.

)1()1()(

n

nbZ

n

naPbXaP

)1(

5.0

)1(

5.0)(

n

nbZ

n

naPbXaP

The Exponential DistributionThe Exponential Distribution

The exponential random variable T (t>0) has a The exponential random variable T (t>0) has a probability density functionprobability density function

Where Where is the mean number of occurrences per is the mean number of occurrences per unit time, t is the number of time units until the unit time, t is the number of time units until the next occurrence, and next occurrence, and ee = 2.71828. . . Then T is = 2.71828. . . Then T is said to follow said to follow an exponential probability distributionan exponential probability distribution. .

The cumulative distribution function isThe cumulative distribution function is

The distribution has mean 1/The distribution has mean 1/ and variance 1/ and variance 1/22

0 for t )( tetf

0 for t 1)( tetF

Probability Density Function for an Probability Density Function for an Exponential Distribution with Exponential Distribution with = 0.2 = 0.2

x

f(x)

Lambda = 0.2

0 10 20

0.0

0.1

0.2

Joint Cumulative Distribution FunctionsJoint Cumulative Distribution Functions

Let XLet X11, X, X22, . . .X, . . .Xkk be continuous random variables be continuous random variables

i.i. Their joint Their joint cumulative distribution functioncumulative distribution function, F(x, F(x11, , xx22, . . .x, . . .xkk) defines the probability that simultaneously X) defines the probability that simultaneously X11 is less than xis less than x11, X, X22 is less than x is less than x22, and so on; that is, and so on; that is

ii.ii. The cumulative distribution functions F(xThe cumulative distribution functions F(x11), ), F(xF(x22), . . .,F(x), . . .,F(xkk) of the individual random variables are ) of the individual random variables are

called their called their marginal distribution functionsmarginal distribution functions. For . For any i, F(xany i, F(xii) is the probability that the random variable X) is the probability that the random variable Xii does not exceed the specific value xdoes not exceed the specific value xii..

iii.iii. The random variables are The random variables are independentindependent if and only if if and only if

)(),,,( 221121 kkk xXxXxXPxxxF

)()()(),,,( 2121 kk xFxFxFxxxF

CovarianceCovariance

Let X and Y be a pair of continuous random Let X and Y be a pair of continuous random variables, with respective means variables, with respective means xx and and yy. . The expected value of (x - The expected value of (x - xx)(Y - )(Y - yy) is called ) is called the the covariancecovariance between X and Y. That is between X and Y. That is

An alternative but equivalent expression can An alternative but equivalent expression can be derived asbe derived as

If the random variables X and Y are If the random variables X and Y are independent, then the covariance between independent, then the covariance between them is 0. However, the converse is not true.them is 0. However, the converse is not true.

)])([(),( yx YXEYXCov

yxXYEYXCov )(),(

CorrelationCorrelation

Let X and Y be jointly distributed random Let X and Y be jointly distributed random variables. The variables. The correlationcorrelation between X and Y is between X and Y is

YX

YXCovYXCorr

),(

),(

Sums of Random VariablesSums of Random Variables

Let XLet X11, X, X22, . . .X, . . .Xkk be k random variables with means be k random variables with means 11, , 22,. . . ,. . . kk and variances and variances 11

22, , 2222,. . ., ,. . ., kk

22. The following . The following properties hold:properties hold:

i.i. The mean of their sum is the sum of their means; that The mean of their sum is the sum of their means; that isis

ii.ii. If the covariance between every pair of these random If the covariance between every pair of these random variables is 0, then the variance of their sum is the variables is 0, then the variance of their sum is the sum of their variances; that issum of their variances; that is

However, if the covariances between pairs of random However, if the covariances between pairs of random variables are not 0, the variance of their sum isvariables are not 0, the variance of their sum is

kkXXXE 2121 )(

222

2121 )( kkXXXVar

),(2)(1

1 1

222

2121 j

K

i

K

ijikk XXCovXXXVar

Differences Between a Pair of Differences Between a Pair of Random VariablesRandom Variables

Let X and Y be a pair of random variables with means Let X and Y be a pair of random variables with means XX and and YY and variances and variances XX

2 2 and and YY22. The following properties . The following properties

hold:hold:

i.i. The mean of their difference is the difference of their The mean of their difference is the difference of their means; that ismeans; that is

ii.ii. If the covariance between X and Y is 0, then the If the covariance between X and Y is 0, then the variance of their difference is variance of their difference is

iii.iii. If the covariance between X and Y is not 0, then the If the covariance between X and Y is not 0, then the variance of their difference isvariance of their difference is

YXYXE )(

22)( YXYXVar

),(2)( 22 YXCovYXVar YX

Linear Combinations of Linear Combinations of Random VariablesRandom Variables

The linear combination of two random variables, X The linear combination of two random variables, X and Y, isand Y, is

Where a and b are constant numbers.Where a and b are constant numbers.The mean for W is,The mean for W is,

The variance for W is,The variance for W is,

Or using the correlation,Or using the correlation,

If both X and Y are joint normally distributed If both X and Y are joint normally distributed random variables then the resulting random random variables then the resulting random variable, W, is also normally distributed with mean variable, W, is also normally distributed with mean and variance derived above.and variance derived above.

bYaXW

YXW babYaXEWE ][][

),(222222 YXabCovba YXW

YXYXW YXabCorrba ),(222222