ENGINEERING MATERIALS CHAPTER THREE.pdf

Materials

1. CHAPTER THREE
Discrete and continuous density functions
Prepared by:
Muhabaw A.

2. Outline
2
❖Discrete probability mass function
❖Continuous density function
❖Mean and Variance
❖Some special distributions
❖Conditionals distribution

3. 3
❑Supposethatthejumpsin�
??????(�)ofadiscreteR.V.Xoccuratthepoints
�
1,�
2,…,wherethesequencemaybeeitherfiniteorcountablyinfinite,and
weassume�
&#3627408470;<&#3627408485;
&#3627408471;if&#3627408470;<&#3627408471;.
❑Thefunction&#3627408477;
??????&#3627408485;iscalledtheprobabilitymassfunction(pmf)ofthe
discreteR.V.X.
Discrete probability mass function (pmf)

4. 4
Cumulativedistributionfunction(cdf)ofpmf:-thecdfof&#3627408441;
??????(&#3627408485;)ofa
discreteR.VXcanbeobtainedby

5. 5
Example3.1:-Givenaprobabilitymassfunctionf(x)=bx
3
forx=1,2,3.Find
thevalueofb.
Solution:Accordingtothepropertiesofprobabilitymassfunction,
෍
??????∈??????
&#3627408467;&#3627408485;=1
෍
??????=1
3
&#3627408467;&#3627408485;=1=&#3627408463;1
3
+2
3
+3
3
&#3627408463;36=1⇒&#3627408463;=
1
36

6. 6
Example 3.2:-The probability mass function example is given below :
Let X be a random variable, and P(X=x) is the PMF given by,
&#3627408485;012345 6 7
&#3627408467;(&#3627408485;)0&#3627408472;2&#3627408472;2&#3627408472;3&#3627408472;&#3627408472;
2
2&#3627408472;
2
7&#3627408472;
2
+&#3627408472;
i.Determine the value of k
ii.Find the probability P(X≤ 6) and P(3<x≤ 6 )

7. Outline
7
❖Discrete probability mass function
❖Continuous probability density function
❖Mean and Variance
❖Some special distributions
❖Conditionals distribution

8. Continuous probability density function(pdf)
8
Let ??????
????????????=
??????&#3627408493;
????????????
????????????
➢Thefunction??????
????????????iscalledtheprobabilitydensityfunction(pdf)ofthe
continuousr.v.X.

9. 9
Example 3.3:-If the probability density function is given as:
&#3627408467;&#3627408485;=ቊ
&#3627408485;&#3627408485;−1,0≤&#3627408485;<3
&#3627408485;, &#3627408485;≥3
The find ??????(1<&#3627408485;<2)?
Solution
??????1<&#3627408485;<2=න
1
2
&#3627408485;&#3627408485;−1&#3627408465;&#3627408485;
Example 3.4:If X is a continuous random variable with the probability
density function given as:
&#3627408467;&#3627408485;=ቐ
&#3627408463;&#3627408466;
−??????
2
,&#3627408485;≥0
0,&#3627408476;&#3627408481;ℎ&#3627408466;&#3627408479;&#3627408484;&#3627408470;&#3627408480;&#3627408466;
Then find the values of &#3627408463;?

10. Cumulative distribution function (cdf) of pdf
10
The cdf &#3627408493;
??????(??????)of a continuous r.v. X can be obtained by
&#3627408493;
????????????=????????????≤??????=׬
−∞
??????
??????
????????????????????????

11. 11
Example 3.5:-The pdf .of a continuous r.v. X is given by
Find the corresponding cdf &#3627408441;
??????(&#3627408485;)and sketch &#3627408467;
??????(&#3627408485;)and &#3627408441;
??????(&#3627408485;)
Solution

12. Outline
12
❖Discrete probability mass function
❖Continuous probability density function
❖Mean and Variance
❖Some special distributions
❖Conditionals distribution

13. Mean and Variance function
13
Mean:-The mean (or expected value) of a r.v. X , denoted by &#3627409159;
??????&#3627408476;&#3627408479;&#3627408440;(??????)is
defined by
Moment:-The &#3627408475;
??????ℎ
moment of a r.v. X is defined by
Note that the mean of X is the first moment of X .

14. Mean and Variance function
14
Variance:-The variance of a r.v. X, denoted by ??????
??????
2
or ??????&#3627408462;&#3627408479;(??????)is defined by
??????
??????
&#3627409360;
=????????????????????????=&#3627408492;??????−&#3627408492;??????
&#3627409360;
=&#3627408492;??????−??????
??????
&#3627409360;
=&#3627408492;??????
&#3627409360;
−&#3627408492;??????
&#3627409360;

15. 15
Example 3.6:-Consider a discrete r.v. X whose pmf is given by
Find the meanand varianceof X.
Solution
??????
??????=&#3627408492;??????=
&#3627409359;
&#3627409361;
−&#3627409359;+&#3627409358;+&#3627409359;=&#3627409358;
??????
??????
&#3627409360;
=????????????????????????=&#3627408492;??????−&#3627408492;??????
&#3627409360;
=&#3627408492;[??????−??????
??????
&#3627409360;
]=&#3627408492;??????
&#3627409360;
=
&#3627409359;
&#3627409361;
−&#3627409359;
&#3627409360;
+&#3627409358;
&#3627409360;
+&#3627409359;
&#3627409360;
=
&#3627409360;
&#3627409361;

16. 16
Example 3.6:-Find the mean and variance of the r.v. X if the pdf of X is
Solution
??????
??????=&#3627408492;??????=න
&#3627409358;
&#3627409359;
??????&#3627409360;??????????????????=
&#3627409360;
&#3627409361;
&#3627408492;??????
&#3627409360;
=න
&#3627409358;
&#3627409359;
??????
&#3627409360;
&#3627409360;??????????????????=
&#3627409359;
&#3627409360;
??????
??????
&#3627409360;
=????????????????????????=&#3627408492;??????
&#3627409360;
−&#3627408492;??????
&#3627409360;
=
&#3627409359;
&#3627409360;
−
&#3627409360;
&#3627409361;
&#3627409360;
=
&#3627409359;
&#3627409359;??????

17. Outline
17
❖Discrete probability mass function
❖Continuous probability density function
❖Mean and Variance
❖Some special distributions
❖Conditionals distribution

18. Some special distributions
18
A.Bernoulli Distribution:
✓A r.v. X is called a Bernoulli r.v. with parameter pif its pmf is given by
✓The cdf &#3627408493;
????????????of the Bernoulli r.v. X is given by
✓The meanand varianceof the Bernoulli r.v. X are

19. 19
Example3.7:Abasketballplayercanshootaballintothebasketwitha
probabilityof0.6.Whatistheprobabilitythathemissestheshot?
Solution:WeknowthatsuccessprobabilityP(X=1)=p=0.6
Thus,probabilityoffailureisP(X=0)=1-p=1-0.6=0.4
Answer:TheprobabilityoffailureoftheBernoullidistributionis0.4
Example 3.8:If a Bernoulli distribution has a parameter 0.45 then find its
mean.
Solution:Mean E[X] = p = 0.45
Example 3.9:If a Bernoulli distribution has a parameter 0.72 then find its
variance.
Solution:Variance Var[X] = p (1-p) = 0.72 (0.28) = 0.2016

20. Some special distributions
20
B.Binomial Distribution:
✓A r.v. X is called a binomial r.v. with parameters (n,p) if its pmf is given b
✓which is known as the binomial coefficient.
✓The corresponding cdf of X

21. 21
Example3.10:-Abagcontains10marbles,6ofwhichareblueand4arered.An
experimentconsistsofpickingamarble(atrandom)fromthebag,makinganoteofits
colorandputtingitbackinthebag.Thisexperimentisrepeated5times.
Whatistheprobabilityofpickingexactly3bluemarbles?
Solution
Since the marble is put back in the bag at the end of each trial, the outcome of
each pick is independent of the previous. There are only two possible outcomes:
✓picking a blue (success),&#3627408477;=
6
10
=0.6
✓not picking a blue (failure),&#3627408478;=
4
10
=0.4
✓WedefinethediscreterandomvariableXasthe"numberofbluemarblespicked".
Usingthebinomialprobabilitydistributionfunctionwith5trials,n=5,probabilityof
asuccessp=0.6,for3successes,r=3,wefind:

22. Some special distributions
22
C.PoissonDistribution:
✓A r.v. X is called a Poisson r.v. with parameter &#3627409158;(>0) if its pmf is given

23. 23
Example3.11:-Thenumberoftelephonecallsarrivingataswitchboardduring
any10-minuteperiodisknowntobeaPoissonr.v.Xwith&#3627409158;=2.
(a)Findtheprobabilitythatmorethanthreecallswillarriveduringany10-
minuteperiod.
(b)Findtheprobabilitythatnocallswillarriveduringany10-minuteperiod
Solution
(a)ThepmfofXis

24. Some special distributions
24
D.UniformDistribution:
✓A r.v. X is called a uniform r.v. over (a, b) if its pdf is given by

25. Some special distributions
25
E.ExponentialDistribution:

26. Outline
26
❖Discrete probability mass function
❖Continuous probability density function
❖Mean and Variance
❖Some special distributions
❖Conditionals distribution

27. Conditionals distribution
27
❑TheconditionalprobabilityofaneventAgiveneventBwithP(B)>0
??????&#3627408436;&#3627408437;=
??????&#3627408436;∩&#3627408437;
??????&#3627408437;
,&#3627408484;ℎ&#3627408466;&#3627408479;&#3627408466;??????&#3627408437;>0.
❑Theconditionalcdf&#3627408441;
??????(&#3627408485;|&#3627408437;)ofar.vX,giveneventBisdefinedby
&#3627408441;
??????&#3627408485;&#3627408437;=????????????≤&#3627408485;&#3627408437;=
????????????≤&#3627408485;∩&#3627408437;
??????&#3627408437;
➢IfXisadiscreter.vX,thentheconditionalpmf&#3627408477;
??????(&#3627408485;|&#3627408437;)isdefinedby
??????
??????????????????=????????????=????????????=
????????????=??????∩??????
????????????
➢IfXisacontinuousr.vX,thentheconditionalpdf&#3627408467;
??????(&#3627408485;|&#3627408437;)isdefinedby
&#3627408467;
??????&#3627408485;&#3627408437;=
&#3627408465;&#3627408441;
??????&#3627408485;&#3627408437;
&#3627408465;&#3627408485;

28. 28