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Chapter 3

Basic Concepts in Statistics

and Probability

3.6 Continuous Distributions

• Normal distribution

• Student t-distribution

• Exponential distribution

• Lognormal distribution

• Weibull distribution

• Extreme value distribution

• Gamma distribution

• Chi-square distribution

• Truncated normal distribution

• Bivariate and multivariate

normal distribution

• F distribution

• Beta distribution

• Uniform distribution

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The normal distribution (also called the Gaussian distribution) is by far the most commonly used distribution in statistics. This distribution provides a good model for many, although not all, continuous populations.

The normal distribution is continuous rather than discrete. The mean of a normal population may have any value, and the variance may have any positive value.

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3.6.1 Normal Distributions

Probability Density Function, Mean,

and Variance of Normal Dist.

The probability density function of a normal

population with mean and variance 2 is

given by

If X ~ N(, 2), then the mean and variance

of X are given by

4

2 2

X

X

2 2( ) / 21( ) ,

2

xf x e x

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68-95-99.7% Rule

This figure represents a plot of the normal probability density

function with mean and standard deviation . Note that the

curve is symmetric about , so that is the median as well as

the mean. It is also the case for the normal population.

• About 68% of the population is in the interval .

• About 95% of the population is in the interval 2.

• About 99.7% of the population is in the interval 3.

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Standard Units

• The proportion of a normal population that is

within a given number of standard deviations of

the mean is the same for any normal population.

• For this reason, when dealing with normal

populations, we often convert from the units in

which the population items were originally

measured to standard units.

• Standard units tell how many standard deviations

an observation is from the population mean.

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Standard Normal Distribution

In general, we convert to standard units by subtracting the mean and dividing by the standard deviation. Thus, if x is an item sampled from a normal population with mean and variance 2, the standard unit equivalent of x is the number z, where

z = (x - )/.

The number z is sometimes called the “z-score” of x. The z-score is an item sampled from a normal population with mean 0 and standard deviation of 1. This normal distribution is called the standard normal distribution.

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Finding Areas Under the Normal

Curve • The proportion of a normal population that lies within a

given interval is equal to the area under the normal probability density above that interval. This would suggest integrating the normal pdf, but this integral does not have a closed form solution.

• So, the areas under the curve are approximated numerically and are available in Table B. This table provides area under the curve for the standard normal density. We can convert any normal into a standard normal so that we can compute areas under the curve.

• The table gives the area in the right-hand tail of the curve between and z. Other areas can be calculated by subtraction or by using the fact that the normal distribution is symmetrical and that the total area under the curve is 1.

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Normal Probabilities

Excel:

NORM.DIST(x, mean, standard_dev, cumulative)

NORM.INV(probability, mean, standard_dev)

NORM.S.DIST(z)

NORM.S.INV(probability)

Minitab:

Calc Probability Distributions Normal

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Linear Functions of Normal

Random Variables

Let X ~ N(, 2) and let a ≠ 0 and b be constants.

Then aX + b ~ N(a + b, a22).

Let X1, X2, …, Xn be independent and normally distributed with means 1, 2,…, n and variances 1

2, 22,…, n

2. Let c1, c2,…, cn be constants, and c1 X1 + c2 X2 +…+ cnXn be a linear combination. Then

c1 X1 + c2 X2 +…+ cnXn

~ N(c11 + c2 2 +…+ cnn, c121

2 + c222

2 + … +cn2n

2)

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Distributions of Functions of Normal

Random Variables

Let X1, X2, …, Xn be independent and normally distributed

with mean and variance 2. Then

Let X and Y be independent, with X ~ N(X, X2) and

Y ~ N(Y, Y2). Then

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2

~ , .σ

X N μn

2 2

2 2

~ ( , )

~ ( , )

X Y X Y

X Y X Y

X Y N μ μ σ σ

X Y N μ μ σ σ

3.6.2 t Distribution

• If X~N(, 2)

• If is Not known, but n30

• Let X1,…,Xn be a small (n < 30) random sample from

a normal population with mean . Then the quantity

has a Student’s t distribution with n -1 degrees of

freedom (denoted by tn-1). 12

n

XZ

/

)(

(3.7)

ns

XZ

/

)( (3.8)

ns

Xt

/

)( (3.9)

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More on Student’s t

• The probability density of the Student’s t

distribution is different for different degrees of

freedom.

• The t curves are more spread out than the

normal.

• Table C, called a t table, provides probabilities

associated with the Student’s t distribution.

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t Distribution

The probability density function of t

distribution is given by

𝑓 𝑡 =1

(𝑛 − 1)𝜋

Γ(𝑛2)

Γ,𝑛 − 12

-(1 +

𝑡2

𝑛 − 1)−𝑛/2

−∞ < 𝑡 < ∞

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t Distribution

www.boost.org/.../graphs/students_t_pdf.png

Other uses of t Distribution

𝑡 =𝜃 − 𝜃

𝑠𝜃

Where is the parameter to be estimated,

𝜃 is the estimator based on a sample,

𝑠𝜃 is the sample estimator of 𝜎𝜃

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3.6.3 Exponential Distribution

• The exponential distribution is a continuous

distribution that is sometimes used to model the

time that elapses before an event occurs (life

testing and reliability).

• The probability density of the exponential

distribution involves a parameter, which is the

mean of the distribution, , whose value

determines the density function’s location and

shape.

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Exponential R.V.:

pdf, cdf, mean and variance

The pdf of an exponential r.v. is

𝑓 𝑥 =1

𝜃𝑒−𝑥/𝜃 𝑥 > 0

The cdf of an exponential r.v. is 𝐹 𝑥 = 1 − 𝑒−𝑥/𝜃 𝑥 > 0

The mean of an exponential r.v. is

μx =

The variance of an exponential r.v. is

σx2 = 2.

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(3.10)

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Exponential Probability Density

Function

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=1/

Exponential Probabilities

Excel:

EXPONDIST(x, lambda, cumulative)

Minitab:

Calc Probability Distributions Exponential

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Example

A radioactive mass emits particles according to a

Poisson process at a mean rate of 15 particles

per minute. At some point, a clock is started.

1. What is the probability that more than 5 seconds

will elapse before the next emission?

2. What is the mean waiting time until the next

particle is emitted?

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Lack of Memory Property

The exponential distribution has a property known

as the lack of memory property:

If T ~ Exp(1/), and t and s are positive

numbers, then

P(T > t + s | T > s) = P(T > t).

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Example

The lifetime of a transistor in a particular circuit has an exponential distribution with mean 1.25 years.

1. Find the probability that the circuit lasts longer than 2 years.

2. Assume the transistor is now three years old and is still functioning. Find the probability that it functions for more than two additional years.

3. Compare the probability computed in 1. and 2.

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3.6.4 Lognormal Distribution

• For data that contain outliers, the normal distribution is generally not appropriate. The lognormal distribution, which is related to the normal distribution, is often a good choice for these data sets.

• If X ~ N(,2), then the random variable Y = eX has the lognormal distribution with parameters and 2.

• If Y has the lognormal distribution with parameters and 2, then the random variable X = lnY has the N(,2) distribution.

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Lognormal pdf, mean, and

variance

The pdf of a lognormal random variable with parameters

and 2 is

𝑓 𝑥 =1

2𝜋𝜎 1

𝑥 𝑒𝑥𝑝 −

1

2𝜎2(ln 𝑥 − 𝜇)2 𝑥 > 0

Mean:

Variance:

2/2

)( eXE

22 222)( eeYV

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Lognormal Probability Density

Function

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=0

=1

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Lognormal Probabilities

Excel:

LOGNORM.DIST(x, mean, standard_dev)

Minitab:

Calc Probability Distributions Lognormal

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Example

When a pesticide comes into contact with the skin,

a certain percentage of it is absorbed. The

percentage that is absorbed during a given time

period is often modeled with a lognormal

distribution. Assume that for a given pesticide,

the amount that is absorbed (in percent) within

two hours is lognormally distributed with a mean

of 1.5 and standard deviation of 0.5. Find the

probability that more than 5% of the pesticide is

absorbed within two hours.

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3.6.5 Weibull Distribution

The Weibull distribution is a continuous random variable that is used in a variety of situations. A common application of the Weibull distribution is to model the lifetimes of components. The Weibull probability density function has two parameters, both positive constants, that determine the scale and shape. We denote these parameters (scale) and (shape).

If = 1, the Weibull distribution is the same as the exponential distribution.

The case where =1 is called the standard Weibull distribution

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Weibull R.V.

The pdf of the Weibull distribution is

𝑓 𝑥 = 𝛼𝛽(𝛼𝑥)𝛽−1𝑒−(𝛼𝑥)𝛽 𝑥 > 0

The mean of the Weibull is

𝜇𝑥 =1

𝛼Γ(1 +

1

𝛽)

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The variance of the Weibull is

𝜎𝑥2 =

1

𝛼2 Γ 1 +

2

𝛽− Γ(1 +

1

𝛽)

2

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Weibull Probability Density

Function

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=1, =5

=0.5, =1 =0.2, =5

Weibull R.V.

The cdf of the Weibull distribution (with =1) is

𝐹 𝑥 = 1 − 𝑒−(𝑥)𝛽 𝑥 > 0

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Weibull Probabilities

Excel:

WEIBULL(x, alpha, beta, cumulative)

Minitab:

Calc Probability Distributions Weibull

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3.6.6 Extreme Value Distribution

Extreme value distributions are often used in reliability work.

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Extreme Value Distribution

The pdf of the Extreme value distribution is

𝑓 𝑥 =1

𝑏exp ,

𝑥 − 𝑎

𝑏-exp *− exp

𝑥 − 𝑎

𝑏+ − ∞ < 𝑥 < ∞

with a the location parameter and b the scale parameter

The cdf of the Extreme value distribution is

𝐹 𝑥 = 1 − exp *−exp𝑥 − 𝑎

𝑏+

• The Weibull distribution is the limiting distribution of the

extreme value distribution

• The natural log of a Weibull random time is an extreme value

random observation. 35

Extreme Value Distribution

36 http://www.itl.nist.gov/div898/handbook/apr/section1/apr163.htm

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3.6.7 Gamma Distribution

The gamma distribution is frequently a probability model for waiting times; for instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution.

Gamma function is defined by

𝑓 𝑥 =1

𝛽𝛼Γ(𝛼)𝑥𝛼−1𝑒−𝑥/𝛽 𝑥 > 0

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(3.11)

Where >0 (shape) and >0 (scale)

Gamma Distribution

• Standard gamma distribution when =1

• Mean: Variance: 2

• If k is an integer, then the distribution represents an Erlang distribution.

• If = 1, the gamma distribution is the same as the exponential.

• The gamma function has the following properties:

1. If r is any integer, then (r) = (r – 1)!.

2. For any r, (r + 1) = r (r).

3. Γ1

2= π

4. Γ 𝑟 = 𝑥𝑟−1𝑒−𝑥𝑑𝑥∞

0

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Gamma Probability Density

Function

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=1, =1

=3, =0.5

=5, =1

Gamma Probabilities

Excel:

GAMMA.DIST(x, alpha, beta, cumulative)

GAMMA.INV(probability, alpha, beta)

GAMMALN(x)

Minitab:

Calc Probability Distributions Gamma

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• If = r/2, and =2 where r is a positive integer, the distribution is called a chi-square distribution with r degrees of freedom.

• The chi-square distribution results when independent variables with standard normal distributions are squared and summed.

• The probability density function of the chi-square distribution is

𝑓 𝑥 =𝑒−𝑥2 𝑥

𝜈2−1

2𝜈2Γ(𝜈

2) 𝑥 ≥ 0

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3.6.8 Chi-Squre Distribution

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Chi-Squre Distribution

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3.6.9 Truncated Normal Distribution

• Left truncated

• Right truncated

• Doubly truncated

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Truncated Normal Distribution

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Left Truncated Normal Distribution

• Left truncated normal distribution

𝑓 𝑥 =𝑘

𝜎 2𝜋𝑒−1/2,(𝑥−𝜇)/𝜎-

2 𝑥 ≥ 𝑎

Where

𝑘 = 1 − Φ(𝑎 − 𝜇

𝜎)−1

Where a is the truncation point, and is the

cumulative area under the normal curve.

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Lift Truncated Normal Distribution

• Let c = (𝑎 − 𝜇 )/𝜎

• Expected value

𝐸 𝑥 = 𝜇 + 𝑘𝜎𝑓,𝑐-

Where

𝑘 = 1 −Φ(𝑐) −1

𝑓 𝑐 =1

2𝜋𝑒−

12𝑐2

• Variance 𝑉𝑎𝑟 𝑥 = 𝜎2*1 + 𝑐𝑓(𝑐),1 − Φ 𝑐 -−1 − ,𝑓(𝑐)-2,1 − Φ 𝑐 -−2+

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(3.12)

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Right Truncated Normal Distribution

• Right truncated normal distribution

𝑓 𝑥 =𝑘′

𝜎 2𝜋𝑒−1/2,(𝑥−𝜇)/𝜎-

2 𝑥 ≥ 𝑎

Where

𝑘′ = Φ(𝑎′ − 𝜇

𝜎)

−1

Where a’ is the truncation point, and is the

cumulative area under the normal curve.

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Right Truncated Normal Distribution

• Let c′ = (𝑎′ − 𝜇 )/𝜎

• Expected value

𝐸 𝑥 = 𝜇 − 𝑘′𝜎𝑓,𝑐′-

Where

𝑘 = Φ(𝑐′) −1

𝑓 𝑐′ =1

2𝜋𝑒−

12𝑐′2

• Variance 𝑉𝑎𝑟 𝑥 = 𝜎2*1 + 𝑐′𝑓(𝑐′),1 − Φ 𝑐′ -−1 − ,𝑓(𝑐′)-2,1 − Φ 𝑐′ -−2+

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3.6.10 Bivariate and Multivariate

Normal Distribution

• Normal distribution

• For 2 independent variables, x1~N(1,12),

x2~N(2,22)

𝑓 𝑥1, 𝑥2 =1

2𝜋𝜎1𝜎2exp,−(𝑥1−𝜇1)

2

2𝜎12 -−(𝑥2−𝜇2)

2

2𝜎22 -

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2 2( ) / 21( ) ,

2

xf x e x

(3.13)

Bivariate Normal Distribution

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Bivariate Normal Distribution

• Bivariate Normal distribution

𝑓 𝑥1, 𝑥2 =1

2𝜋𝜎1𝜎2exp,−(𝑥1−𝜇1)

2

2𝜎12 -−(𝑥2−𝜇2)

2

2𝜎22 -

• Let 𝕩 =𝑥1𝑥2

, 𝕞 =𝜇1𝜇2

, 𝕊 =𝜎1

2 0

0 𝜎22 ,

𝑓 𝕩 =1

2𝜋|𝕊|1/2exp,−1

2(𝕩−𝕞)′𝕊−1(𝕩−𝕞)-

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(3.13)

Multivariate Normal Distribution

• Let 𝕩 =

𝑥1⋮𝑥𝑝

, 𝕞 =

𝜇1⋮𝜇𝑝

, 𝕊 =𝜎1

2 ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝜎𝑝

2

𝑓 𝕩 =1

(2𝜋)𝑝/2|𝕊|1/2exp,−1

2(𝕩−𝕞)′𝕊−1(𝕩−𝕞)-

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(3.14)

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3.6.11 F Distribution

• Let W, and Y be independent 2 random variables with u and degrees of freedom, the ratio F = (W/u)/(Y/ ) has F distribution.

• Probability Density Function, with u, degrees of freedom,

• Mean

• Variance

x

xuu

xuu

xfu

uu

0

1)()2

()2

(

))(2

()(

2

122

2)2(

)(

forxE

4)4()2(

)2(2)(

2

2

for

u

uxV

F Distribution

• Table V in Appendix A.

u

uf

f,,

,,1

1

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3.6.12 Beta Distribution

Beta function is defined by

𝑓 𝑥 =𝑟+𝑠−1 !

𝑟−1 ! 𝑠−1 !𝑥𝑟−1(1 − 𝑥)𝑠−1 0 ≤ 𝑥 ≤ 1

Mean of Beta distribution: r/(r+s)

Variance of Beta distribution: rs(r+s)-2(r+s+1)-1

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Where r and s are shape parameters of Beta distribution

3.6.12 Beta Distribution

56 http://astarmathsandphysics.com/university_maths_notes/probability_and_statistics/probability_

and_statistics_the_beta_distribution.html

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3.6.12 Beta Distribution

Beta distribution and F distribution

𝐵 𝛼; 𝑟, 𝑠 =2𝑟/2𝑠)𝐹(𝛼;2𝑟,2𝑠

1+(2𝑟/2𝑠)𝐹(𝛼;2𝑟,2𝑠)

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Where B(; r, s) denotes the (1- ) percentile of a beta distribution

with parameters r and s.

3.6.13 Uniform Distributions

The uniform distribution has two parameters, a

and b, with a < b. If X is a random variable with

the continuous uniform distribution then it is

uniformly distributed on the interval (a, b). We

write X ~ U(a,b).

The pdf is

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1,

( )

0, otherwise

a x bf x b a

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Uniform Distribution:

Mean and Variance

If X ~ U(a, b).

Then the mean is

and the variance is

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2X

a bμ

22 ( )

.12

X

b aσ