Upload
knowledge-creators
View
233
Download
0
Embed Size (px)
Citation preview
8/14/2019 Theoretical Distribution Statistics)
1/22
5 DECEMBER 09 www.afterschool.tk 1
THEORETICAL DISTRIBUTION(STATISTICS)
by :
DR. T.K. JAIN
AFTERSCHOOL
centre for social entrepreneurship
sivakamu veterinary hospital road
bikaner 334001 rajasthan, indiaFOR PGPSE PARTICIPANTS
mobile : 91+9414430763
8/14/2019 Theoretical Distribution Statistics)
2/22
5 DECEMBER 09 www.afterschool.tk 2
WHAT IS IT?
When we can allocate probability for differentfrequency points/ class intervals, out of the
total frequency, it is called theoreticaldistribution. It exists only in theory. The
observed (actual) frequency distribution maybe similar to it. This helps us in prediction,
projections, and in decision making. Forexample: we may say that out of 1000 toss ofcoins, we will get 500 heads (actually, it may
not be 500).
8/14/2019 Theoretical Distribution Statistics)
3/22
5 DECEMBER 09 www.afterschool.tk 3
Uses of theoretical distribution
It is used in following purpose :
statistical analysisprediction of future -
prediction of the data on the basis of the the
distributiondata analysis
8/14/2019 Theoretical Distribution Statistics)
4/22
5 DECEMBER 09 www.afterschool.tk 4
Characteristics
Theoretical distribution contains all thecharacteristics of population as it is based on
population data. So it has all the characteristics
of population parameters
8/14/2019 Theoretical Distribution Statistics)
5/22
5 DECEMBER 09 www.afterschool.tk 5
Types of distribution
There are two types of distributions :
discreet individual values for example :1,2,3,4
continuous continuous values for example : 1to 2, 2 to 3, 3 to 4 etc.
8/14/2019 Theoretical Distribution Statistics)
6/22
5 DECEMBER 09 www.afterschool.tk 6
Discrete distributions
Important ones are :
binomial
poisson
8/14/2019 Theoretical Distribution Statistics)
7/22
5 DECEMBER 09 www.afterschool.tk 7
Continuous distributions
Important ones are :
1. normal
2. chi-square
3. student (or T distribution)
4. F distribution
8/14/2019 Theoretical Distribution Statistics)
8/22
5 DECEMBER 09 www.afterschool.tk 8
Binomial distribution
When only two outcomes are possible,binomial distribution is prepared example
throw of coins (head or tail)trials are independent
P = probability of success
Q= probability of failurefinite number of trials with exclusive and
exhaustive outcomes
8/14/2019 Theoretical Distribution Statistics)
9/22
5 DECEMBER 09 www.afterschool.tk 9
Properties of binomial distribution
Mean = N*P
example : throw coin 100 timesN=100, P=.5, thus mean is 100*.5 = 50
variance =N*p*q
for coins : 100*.5*.5 = 25standard distribution = sqrt(25) = 5
8/14/2019 Theoretical Distribution Statistics)
10/22
5 DECEMBER 09 www.afterschool.tk 10
Ex: a coin is tossed 8 times, whatis the probability of getting 4
heads?
Formula of probability mass function ofbionmial distribution : Ncr*p^r*q^(n-r)
N=8, P=.5, Q=.5, r=4
8C4*(.5)^4*(.5)^4
=105/512
8/14/2019 Theoretical Distribution Statistics)
11/22
5 DECEMBER 09 www.afterschool.tk 11
Ex: a coin is tossed 8 times, whatis the probability of getting at
least 4 heads?
Formula of probability mass function of
bionmial distribution : Ncr*p^r*q^(n-r)N=8, P=.5, Q=.5, r=4,5,6,7,8
when r is 4, =8C4*(.5)^4*(.5)^4
when r is 5, =8C4*(.5)^5*(.5)^3so answer : = 837/1024
8/14/2019 Theoretical Distribution Statistics)
12/22
5 DECEMBER 09 www.afterschool.tk 12
What is the probability of getting2 sundays out of 15 days selected
at random?
N = 15, R = 2, P = 1/7, because only 1 Sundayin a week Q = 6/7 (all other days )
=15C2*(1/7)^2*(6/7)^13
8/14/2019 Theoretical Distribution Statistics)
13/22
5 DECEMBER 09 www.afterschool.tk 13
Poisson distribution
We use it when the possibility of success is in avery small time interval is given. There is very
little possibilty of getting 2 successes. It isgenerally used for situations like queue etc.
Formula : e^kt*(kt)^x / x!
Where e=2.71828 k = chance of success(between 0 to 1)
t = time and x is the value that we want to test.
8/14/2019 Theoretical Distribution Statistics)
14/22
5 DECEMBER 09 www.afterschool.tk 14
Find the mean and standard deviation of Xwhere X is poisson variate satisfying the
conditions P(x=2) = P (x=3) Options : mean=3,4,5 standard deviations : sqrt (2), sqrt (3), 2
e^kt*(kt)^x / x!
2.71828^kt*(kt)^3 / 3! = 2.71828^kt*(kt)^2/2!
Kt^3/6 = kt^2/2
kt^3 = kt^2*3
thus kt = mean = 3
standard deviation = square root of mean (in poisson
distribution) so standard deviation is sqrt(3)
8/14/2019 Theoretical Distribution Statistics)
15/22
5 DECEMBER 09 www.afterschool.tk 15
Between 5 to 6 PM the average number ofphone clls per minute is 4. what is the
probability that there is no phone callduring a minute?
e^kt*(kt)^x / x!
E = 2.71828, t = 1 minute, k = 4, x = 0 (nophone call)
solving we get : = .018 answer
8/14/2019 Theoretical Distribution Statistics)
16/22
5 DECEMBER 09 www.afterschool.tk 16
2% Cars are defective. What is theprobability that out of 150 cars,
there is exactly one defective car?
E =2.71828, KT = 2% of 150 = 3
x = 1.
solving the formula we get : .15 answer
8/14/2019 Theoretical Distribution Statistics)
17/22
5 DECEMBER 09 www.afterschool.tk 17
Normal distribution
Developed by Karl Gauss
perfectly symmetrical, mean=mode=medianvalues = equally divided from mean
+ or one sigma = .68
+ or 2 sigma = .9546+ or 3 sigma = .9973 _
8/14/2019 Theoretical Distribution Statistics)
18/22
5 DECEMBER 09 www.afterschool.tk 18
Out of 500 workers mean and standarddeviation of wages are 500 and 48
respectively. What is the number ofworkers with more than 600 as wages.
Here 500 is mid point, so it represents .5
we are interested in 600, which is more than 500
600 500 / 48 = 100/48 = 2.083
now find the value 2.083 in the table of normal distribution,which is .4812 add .5 to it : .9812,
so answer : 1 - .9812 = .0188 or 1.8% or 9 workers. Answer.
8/14/2019 Theoretical Distribution Statistics)
19/22
5 DECEMBER 09 www.afterschool.tk 19
What is the standard deviation ofthe number of recoveries among
48 patients, the probability ofrecovering is .75? Options :
36,81,9,3
Formula : N*P*Q
= 48*.75*.25 = 9 answer
8/14/2019 Theoretical Distribution Statistics)
20/22
5 DECEMBER 09 www.afterschool.tk 20
The incidence of occupational disease in anindustry is such that the workman have a 10%
chance of suffering from it. What is the
probability that out of 5 workmen 3 willcontract the disease?
Formula : NCR*P^r*Q^(n-r)5C3*(.1)^3*(.9)^2
10*.001*.81
= .0081 answer
8/14/2019 Theoretical Distribution Statistics)
21/22
5 DECEMBER 09 www.afterschool.tk 21
What is the probability of success of thebinomial distribution satisfying the
following condition: 4P(X=4) = P(X=2)and having other parameter as 6? options :
.25, .33, .4, .5
Here N = 6 (as this is the only parameter missing).
4* 6C4(p)^4*(q)^2 = 6C2*(p)^2*(q)^4
try with options, .33 gives answer so P = .33
8/14/2019 Theoretical Distribution Statistics)
22/22
5 DECEMBER 09 www.afterschool.tk 22
THANKS....
GIVE YOUR SUGGESTIONS AND JOINAFTERSCHOOOL NETWORK / START
AFTERSCHOOOL SOCIALENTREPRENEURSHIP NETWORK INYOUR CITY
PGPSE WORLD'S MOSTCOMPREHENSIVE PROGRAMME IN
SOCIAL ENTREPRENEURSHIP
mailto:[email protected]:[email protected]