Post on 11-Mar-2023
Recall: Mo*va*on Probability distribu.on (represented by tabular form, formula or graphs)
à behavior of random variable
In general, observa.ons generated by different experiment have similar type of behavior. In prac.ce, several life random phenomenon can be described by some commonly used distribu.on In this lesson, we present these commonly used CONTINUOUS distribu.ons .
Engineering Statistics
Probability Distribu*ons: Con*nuous
Engineering Statistics Engineering Statistics
http://science.utm.my/norhaiza/
Recall Random Variable
Discrete Con.nuous
Probability Distribu.on
Probability
Mass Func*on
Probability Distribu.on
Probability Density
Func*on
Special Probability Distribu*on
Special Probability Distribu*on
Normal Distribu*ons Characteris*cs (draw basic normal)
• the most widely used probability distribu.on in sta.s.cs and an important tool in analysis • also called the Gaussian Distribu*on
Normal Distribu*ons r.v X with p.d.f is a normal r.v with parameters mean and variance such that and è
∞<<∞=−
−
xexfx
- ,21)( 2
2
2)(
σµ
σπ
),(~ 2σµNX
∞<<∞− µ 0>σµ 2σ
Normal Distribu*ons Characteris*cs
• Bell Shape Curve and is Symmetric • Symmetric around the mean: Two halves of the curve are the same (mirror images) • sketch
Normal Distribu*ons Illustra*ons
•
• Effect of different parameter values on the bell-‐shaped normal distribu.on graph
Normal Distribu*ons Finding probabili*es
•
è
• Very difficult to evaluate. What to do? • è need to transform • è work with standard normal variable
?)(),,(~ If 2
=≤≤ bXaPNX σµ
∫−
−
=≤≤b
a
x
dxebXaP 2
2
2)(
21)( σ
µ
σπ
ZX
Normal Distribu*ons Standard Normal Distribu*on
• Transform to i.e. by STANDARDIZING
è
Then
σµ
σµ−
=XZNX and ),,(~ If 2
ZX
σµ−
=XZ
X
)1,0(~ NZ
• Given
• Note: For con*nuous r.v: no significant difference in equality signs
1.0)(7.0)(1.0)(7.0)(1.0)0(
=>
=>
=<
=<
=<<
zZPzZPzZPzZPzZP
)1,0(~ NZ
Finding probabili*es : standard Normal Distribu*on
sketch the area of probability
Finding probabili*es : standard Normal Distribu*on Cumula*ve Distribu*on Func*on for Z,
•
• S.ll difficult to evaluate. • Refer tables
∫∞−
−=≤=Φ
z zdzezZPz
2
21
21)()(π
)(zΦ
Finding probabili*es: Standard Normal Probability Tables
• We can evaluate the probability associated to a standard normal distribu.on either using a scien.fic calculator/computer applica.on such as MicrosoX Excel or sta.s.cal tables. • Sta.s.cal table –many versions ß H.Lee stats table (given z value, prob.?) ß H.Lee stats table (given prob. , z value?)
Note: different books use different area of the curve to calculate std.normal probability dn. Montgomery et al. Cumula.ve standard normal dn. (given z value, prob.?) )( zZP ≤
)0( zZP ≤≤
α=≥ )( zZP α
Appendix A. Finding probabili*es: Standard Normal Probability Tables (Lee)
?)0( =<< zZP α=≥ )( zZP
Given z-‐value, Probability?
Given probability, z-‐value?
α
z 0.00 .. .. 0.09
0.0 0.0000 .. .. 0.0359
. .
. .
. .
1.5 0.4332 .. .. 0.4441
Appendix A. Finding probabili*es: Cumula*ve Standard Normal Distribu*on Tables (Lee), P(0<Z <z)
)0( zZP <<
z0
)59.10( << ZP
59.10
Cumula*ve Standard Normal Dn. Probability Tables,
If Z~ N(0,1), find from tables (sketch it!) 0.9162
=≤ )377.1( (a) ZP
=−> )377.1( (b) ZP = 0.9162
=> )38.1( (d) ZP=−< )38.1( (c) ZP
0.5+P(0 < Z <1.38) =
0838.0 0.5−P(0 < Z <1.38) = 0.0838
Excel note: P(Z<z) =normsdist(1.377)=0.9157 =normsdist(1.38)=0.9162
If Z~ N(0,1), find from tables (sketch it!)
)43.1()43.1()433.1|(|)()43.143.1()433.1|(|)(
)6.04.1()()865.1696.2()()751.1345.0((a)
−<+>=>
<<−=<
=−<<−
=<<−
=≤<
ZPZPZPeZPZPd
ZPcZPbZP 32311.0)35.00()75.10( =<<−<< ZPZP
9658.01935.0
Cumula*ve Standard Normal Dn. Probability Tables
Cumula*ve Standard Normal Dn. Probability Tables
If Z~ N(0,1), find the value of a if
9.0)|(|)(9693.0)()(0793.0)()(7818.0)()(
3802.0)( (a)
=<
=>
=<
=>
=>
aZPeaZPdaZPcaZPbaZP
Excel note:P(Z<a) P(Z>a)=0.3802 1-‐P(Z<a)=0.3802 P(Z<a)=0.6198 à =normsinv(0.6198) =normsinv(0.3802)=0.3805
Finding Probabili*es
If X~ N(300,25), find
=> )305( (a) XP
=≤ )291( (b) XP
=< )312( (c) XP
=> )286( (d) XP
Excel note: P(X<x) =normdist(1.377)=0.9157 =normsdist(1.38)=0.9162
Normal Distribu*ons: Example 2 Suppose the current measurements in a strip of wire are assumed to follow a normal distribu.on with a mean of 10 milliamperes and a variance of 4 mA2. (a) What is the probability that a current measurement
will exceed 13 milliamperes? (0.067) (b) What is the probability that a current measurement is
between 9 and 11 milliamperes? (0.383) (c) Determine the value for which the probability that a
current measurement is below this value is 0.98? (14.1mA)
Exercise(Montgomery, 4th ed. Ex.4-‐57) The speed of a file transfer from a server on campus to a personal
computer at a student’s home on a weekday evening is normally distributed with a mean of 60 kilobits per second and a standard devia.on of 4 kilobits per second. (a) What is the probability that the file will transfer at a speed of 70 kilobits per second or more? (b) What is the probability that the file will transfer at a speed of less than 58 kilobits per second? (c) If the file is 1 megabyte, what is the average .me it will take to transfer the file? (Assume eight bits per byte.)
Recall: Poisson Process
• discrete distribu.on • Poisson dn.à r.v • compute probability of specific number of events during an interval (.me or space)
è The interval, (eg. the .me period/span of space) can be a random variable!
Exponen*al Distribu*on
• rela.onship to Poisson process • Example Ø *me between arrivals at service facili.es Ø *me to failure of components
• Important role in queuing theory & reliability problems
Example 3: Exponen*al Distribu*on
• r.v Y , number of flaws along a length of copper wire • No. of flawà Poisson event • r.v X, length from any point on the copper wire un.l a flaw is detected distribu.on of X depends on distribu.on of Y è In general, • r.v Y , number of flaws along x millimeters of wire • if = mean number of flaw per millimeter, then • r.v X, length from any point on the wire un.l a flaw is detected
λ
∴
X ~ Exp(λ)
Exponen*al Distribu*ons Defini*on
A Con$nuous r.v X having a prob. density func$on where Noteè
xexf λλ −=)(0for ≥x
)()|()(1)(1)()(
2121 tXPtXttXPexFxXPexFxXP
x
x
<=>+<
=−=>
−==<−
−
λ
λ
constant positive a is λ
Lack of memory property
Exponen*al Distribu*ons: Example 4
In a large corporate computer network, user log-‐ons to the system can be modeled as a Poisson process with a mean of 25 log-‐ons per hour. (a)What is the probability that there are no logons in an interval of 6 minutes? (b) What is the probability that the .me un.l the next log-‐on is between 2 and 3 minutes? X, .me in hours from the start of the interval un.l the first log-‐on X~exp(25)
082.025)1.0(0.1
25 ==> ∫∞
− dxeXP x
152.025)05.0033.0(05.0
0.033
25 ==<< ∫ − dxeXP x
Exponen*al Distribu*ons: Example 5: Lack of memory property
Let X denote the .me between detec.ons of a par.cle with a geiger counter and assume that X has an exponen.al distribu.on with mean, 1.4 minutes. What is the probability that we detect a par.cle within 30 seconds of star.ng the counter X, .me to detect a par.cle aXer star.ng the counter X~exp(1/1.4) suppose we turn on the geiger counter and wait 3 minutes without detec*ng a par*cle. What is the probability that a par.cle is detected in the WITHIN THE next 30 seconds?*
3.01)5.0()5.0( 4.15.0
=−==<−
eFXP
3.0)3(1)3()5.3(
)3()5.33()3|5.3( =
−
−=
>
<<=><
FFF
XPXPXXP
Exponen*al Distribu*ons: Exercise (crawshaw,eg.5.22,p.309)
The life.me in years of a TV tube is a r.v T and its p.d.f , f(t) is given by Obtain A in terms of k (a) If the manufacturer finds that out of 1000 such tubes, 371 failed
within the first two years of use, es.mate the value of k. (0.232,3sf) (b) Using the value of k in ques.on (a) correct to 3 significant figures,
calculate the mean and variance of T, giving answers correct to 2 significant figures. (4.3;1.9)
(c) If two such tubes are bought, what is the probability that one fails within its first year and the other lasts longer than six years? (0.103)
Group submission
otherwise , 0 0,t0for ,)(
=
>∞≤≤= − kAetf kt
Other Distribu*ons: Erlang & Gamma Weibull Log-‐normal
• Strictly non-‐nega.ve distribu.ons: posi.ve skewed
38 Fundamental Topics
1.4.8 Other continuous distributions
Other distributions for continuous random variables include Erlang, Gamma, Weibulland log-normal distributions. Unlike normal distribution, these distributions assumethat the variables are strictly non-negative. The list of probability density functionsfor these distributions are listed below:
Distributions Probability Density Functions Mean & Variance
1. Erlang f(x) =µrxr�1e�µx
(r � 1)!for x > 0. E(X) = r
µ
and r = 1, 2, . . .
Note: If r = 1, then Erlang is simply V ar(X) = r
µ
2
an exponential distribution.
2. Gamma f(x) =µrxr�1e�µx
(r � 1)!for x > 0 and r > 0. E(X) = r
µ
Note: If r is an integer, then Gamma is V ar(X) = r
µ
2
simply an Erlang distribution.
3. Weibull f(x) =�
�
⇣x�
⌘��1
exp
�⇣x�
⌘�
�
for x > 0, � > 0 and � > 0. E(X) = ��
✓1 +
1
�
◆
Note: � and � are the shape and V ar(X) = �2�
✓1 +
2
�
◆�
the scale parameters respectively.
If � = 1 then Weibull is simply �2�
✓1 +
1
�
◆�2
an exponential distribution with µ = 1/�.
4. Log-normal f(x) =1
x!p2⇡
exp
� (lnx� ✓)2
2!2
�E(X) = e✓+!
2/2
for 0 < x < 1 and X = exp(W ) V ar(X) = e2✓+!
2(e!
2 � 1)
where W ⇠ N(✓,!2).
The shape of the above distributions for varying values of their parameters can be
investigated via computer software such as Matlab. Further information and examples
for these distributions can be found from Montgomery & Runger (2006).
• Strictly non-‐nega.ve distribu.ons
• Similarly different values its parameters affect the shape of the distribu$ons
Figure 1. Univariate distribution relationships.
The American Statistician, February 2008, Vol. 62, No. 1 47
LEEMIS, Lawrence M.; Jacquelyn T. MCQUESTON (February 2008). "Univariate Distribu.on Rela.onships" (PDF). American Sta.s.cian. 62 (1): 45–53. doi:10.1198/000313008x270448
Rela.onships between
distribu.ons
Simplified rela.onships between distribu.ons htps://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Rela.onships_among_some_of_univariate_probability_distribu.ons.jpg/1920px-‐Rela.onships_among_some_of_univariate_probability_distribu.ons.jpg