qt1 - 05 - probability distributions
TRANSCRIPT
Probability Distributions
Probability Distributions
QUANTTECHINTEUQIASEVIT10S
Contents
Probability Distributions
Random Variable
Expected Value of a Random Variable
Bernoulli Random Variable
Bernoulli Distribution
Binomial Random Variable
Binomial Distribution
Poisson Random Variable
Poission Distribution
How can we predict the length of queues at retail counter ?
Frequency Distributions (Again!)
The results of two experiments are given here
To test hearing ability, words are read out and the listener has to repeat what he has heard
People desiring to join the army are tested among other things on the size of the test
From Frequency Distributions
to Probability Distributions
Actual frequency observed in a group of army recruits
Is it possible for us to define a mathematical expression
that will allow us to predict the relative frequency distribution ?
Without actual experiments ?
If YES then that mathematical expression is a probability distribution
Probability Distribution
Frequency Distribution
Is a listing of the observed frequencies of all the outcomes of an experiment that did occur when the experiment was performed
Probability Distribution
Is a listing of the probabilities of all the possible outcomes that could occur if the experiment was performed
Discrete Probability Distribution
Can take on a finite set of values that can be listed
Size of chest
Score in hearing test
Continuous Probability Distribution
Can take on any set of values within a certain range
Wealth level in a population
Protein content in eggs
Prediction
Can we predict the relative frequency or probability of
Acetic Acid in any collection of eggs ?
Billionaires in any year
Based on the data given here ?
YES : If you can find the probability distribution
Models !
Basic Modelling Technique
Toss a Coin; Head or Tail
How many Heads in 10 Coin Tosses
Chest Size of Army Recruits
Amount of Protein in an Egg
Number of Membersin a Family
Money spent by afamily at multiplex
Number of Defectsper thousand
Percentage of Peoplewho die before 30
Experimental Data or observation
Can be represented by aset of random numbers
That obeys certainrules ..
1 or 0
Integer between 0 and 10
Integer between 30 and 50
Any number between 3 and 5
Integer between0 and 10
Any positive numberless than 2000
Integer between 0 and 5
Any number from0 and 100
Generatedthrough anartificial processthat is either physicalor digitaland then modifiedas percertain mathematical equations
Random Variable
A variable is random if it takes on different values as a result of the outcomes of a genuinely random experiment
A random variable can be
Discrete : takes on a fixed set of values
Continuous : takes on any value in a range
Can be thought of as a value or magnitude that changes from occurrence to occurrence without any predictable sequence
While meeting some necessary conditions
Expected Value
Expected value is fundamental idea in the study of probability distributions
A RV X can take on values x1,x2,x3 ...... xn
Expected Value
Summation of ( Value of RV x Probability of occurrence of this value) across all possible value of the Random Variable
S xi P(X = xi )
The expected value of a RV is equivalent to the mean of the population / sample that it is representing or modelling
So is the case with the standard deviation
Expected Value / Mean
Tossing Coins
Bernoulli Random Variable
Consider a variable R that can take on only two values : 0 and 1
An experiment can result in only two outcomes, A and B
R = 1 when Event A happens
R = 0 when Event B happens
The probability for the two events are as follows
P(A) = P(R=1) = p Binomial => Poisson
Similarity of Processes
Bernoulli Process
Single event with success probability p
Binomial Process
Number of successes in n Bernoulli trials each with success probability p
Mean number of successes = m = np
Poisson Process
Number of arrivals in n time slots ( n is very high > 30) where the probability of arrival in a single time slot is p
Mean number of arrivals = l = np
Poisson Distribution
in terms of average number of events
Probability of exactly x arrivals in time interval T
lx . e-l
P(x) =
x!
Where l = average number of arrivals in interval T
Assumption
The average ( mean ) number of events / unit time can be estimated from past data
Events are NOT simultaneous
There will be some small gap between events (t)
Events are independent of each other
Events are equally likely over the entire time unit
p is constant
Not always true
Poisson @ childbirth
Probability of exactly x childbirths in one day
lx . e-l
P(x) =
x!
Where l = average number of childbirths in a day = 5
P(x=0) = 0.00674
P(x=1) = 0.03370
P(x=2) = 0.08425
P(x=3) = 0.14042
P(x=4) = 0.17552
What is the probability of three or fewer babies ?
What is the probability of at least 4 babies in a day ?
Two Hospitals
Probability of exactly x childbirths in one day
lx . e-l
P(x) =
x!
Where l = average number of childbirths in a day
Probability of exactly x childbirths in one day
(l1+l2)x . e-(l1+l2)
P(x) =
x!
Where l1, l2 = average number of births in the two hospitals
Poisson Distribution
Probability of exactly x arrivals in interval 0->T
lx . e-l
P(x) =
x!
Where l = average number of arrivals in interval [0 to T]
Probability of exactly x events in interval 0-> t
l0x . e-l0
P(x) =
x!
Where l0 = average number of arrivals in interval [0 to t]
l0 = l/n = lt/T
l0 = (l/T)t = lrt
lr = rate at which arrivals happen !
Poisson Distribution
Probability of exactly x arrivals in interval 0-> t
l0x . e-l0
P(x) =
x!
Where l0 = average number of arrivals in interval [0 to t]
l0 = l/n = lt/T
l0 = (l/T)t = lrt
lr = rate at which arrivals happen !
Probability of exactly x arrivals in interval 0-> t
(lrt)x . e-(lrt)
P(x) =
x!
lr = number of arrivals per unit time
lrt = mean number of arrivals in interval t
Exponential Distribution
Interval between two Arrivals
Number of arrivals is modelled by Poisson Distribution
Probability of exactly x arrivals in interval 0-> t
(lrt)x . e-(lrt)
P(x) =
x!
lr = number of arrivals per unit time
lrt = mean number of arrivals in interval t
Interval between two arrivals are modelled by Exponential Distribution
Probability of time t units between two arrivals
P(t) = lr . e-(lrt)
Mean of t = 1/lr
Close Cousins in Randomness
Poisson Variable
Discrete Random Variable
Takes Integer Values
Represents number of events ( or arrivals) over a period of time
Mean Number = lr
Rate of arrival
Number of arrivals per unit time
(lrt)x . e-(lrt)
P(x) =
x!
Exponential Variable
Continuous Random Variable
Takes any positive value, not necessarily integer
Represents the interval between two events ( or arrivals)
Mean interval = 1/lr
P(t) = lr . e-(lrt)
Modelling Queues
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prithwis mukerjee
Probability of Success of Bernoulli Event0.4Bernoulli Event12345678910
Primary Random Number.847.241.634.236.062.255.465.562.310.790
Bernoulli Random Variable0101110010
???Page ??? (???)21/06/2008, 10:03:43Page / Probability of Success of Bernoulli Event0.80101111110
1110101010
1110111001
0111111111
1011100111
1110110111
0110111110
1111111011
1111111111
0111110111
???Page ??? (???)15/06/2008, 21:49:51Page / n7n7n7
p0.1p0.4p0.7
00.4800.0300
10.3710.1310
20.1220.2620.03
30.0230.2930.1
4040.1940.23
5050.0850.32
6060.0260.25
707070.08
???Page ??? (???)16/06/2008, 15:35:42Page / Random VariableProbabilityColumn C
00.4782969
10.3720087
20.1240029
30.0229635
40.0025515
50.0001701
60.0000063
70.0000001
Random VariableProbabilityColumn F
00.0279936
10.1306368
20.2612736
30.290304
40.193536
50.0774144
60.0172032
70.0016384
Random VariableProbabilityColumn I
00.0002187
10.0035721
20.0250047
30.0972404999999999
40.2268945
50.3176523
60.2470629
70.0823543
n21n7p0.4p0.40000.031010.132020.2630.0130.2940.0340.1950.0650.0860.160.0270.157080.1790.17100.13110.09120.05130.02140.01150160170180190200210
???Page ??? (???)16/06/2008, 15:35:42Page / Random VariableProbabilityColumn C
00.0000219369506403778
10.00030711730896529
20.00204744872643526
30.00864478351161556
40.0259343505348467
50.0587845278789858
60.104505827340419
70.149294039057742
80.174176378900699
90.167725401904377
100.134180321523501
110.0894535476823342
120.0496964153790746
130.0229368070980344
140.00873783127544168
150.00271843639680408
160.00067960909920102
170.000133256686117847
180.0000197417312767181
190.00000207807697649664
200.000000138538465099776
210.000000004398046511104
Random VariableProbabilityColumn J
00.0279936
10.1306368
20.2612736
30.290304
40.193536
50.0774144
60.0172032
70.0016384
Shirts SoldMargin / shirt200.000Total MarginXP(X = x)X * P(X=x)MP(M=m)M * P(M=m)
00.0050.00000.0050.00
10.0550.0552000.05511.00
20.1200.2404000.12048.00
30.2000.6006000.200120.00
40.2000.8008000.200160.00
50.2001.00010000.200200.00
60.1000.60012000.100120.00
70.0800.56014000.080112.00
80.0250.20016000.02540.00
90.0140.12618000.01425.20
100.0010.01020000.0012.00
1.0004.1911.000838.20
???Page ??? (???)29/07/2008, 12:05:20Page /