qt1 - 05 - probability distributions

Probability Distributions


QUANTTECHINTEUQIASEVIT10S

Contents


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


Any number between 3 and 5

Integer between0 and 10

Any positive numberless than 2000


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


lx . e-l

P(x) =

x!

Where l = average number of childbirths in a day


(l1+l2)x . e-(l1+l2)

P(x) =

x!

Where l1, l2 = average number of births in the two hospitals


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 !


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 !


(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


(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

Click to edit the title text format

Click to edit the outline text format

Second Outline Level

Third Outline Level

Fourth Outline Level

Fifth Outline Level

Sixth Outline Level

Seventh Outline Level

Eighth Outline Level

Ninth Outline Level

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 /

qt1 - 05 - probability distributions

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