LECTURE 14 SIMULATION AND MODELING Md. Tanvir Al Amin, Lecturer, Dept. of CSE, BUET CSE 411
Feb 23, 2016
LECTURE 14SIMULATION AND MODELINGMd. Tanvir Al Amin, Lecturer, Dept. of CSE, BUETCSE
411
Discrete Uniform Distribution Uniform distribution inside a interval.
Say a random variable is equally likely to take value between i and j inclusive
What is the probability that X = x where
Mean
Variance
jxi
11)(
ij
xp
2)( ji
121)1( 2 ij
Discrete Uniform Distribution
Probability Mass Probability Distribution
otherwise ,0
},.....1,{ x if ,1
1)( jii
ijxP
jx
jxiijix
ix
xF
if , 1
if ,11
if , 0
)(
Binomial Distribution Number of successes in n independent
Bernoulli trials with probability p of success in each trial
Relation between bernoulli and binomial : Suppose a two-tailed experiment
Pick a ball from the urn : Ball is either blue or red So two tailed test Pr { Blue } = 6/10 = 0.6 Pr { Red } = 4/10 = 0.4 This is a bernoulli trial
Binomial Distribution Now suppose we have n such urns…
Pick a ball from urn 1, Pick a ball from urn 2, Pick a ball from urn 3…
All are independent events. Each of these parallel experiments have Pr{red}=0.4, and Pr{blue} = 0.6
Urn 1 Urn 2 Urn 3 Urn 4 Urn 5
Binomial Distribution All are independent events. Each of
these parallel experiments have Pr{red}=0.4, and Pr{blue} = 0.6
Does it mean, all of these experiments will have same outcome ?
NO !!!
One Experiment
2 red, 3 blue balls …
Another Experiment
4 red, 1 blue balls …
Binomial Distribution What is the probability that outcome is 1
red ball ? i.e. (4 blue balls) What is the probability that outcome is 3
red balls ? (and hence 2 blue balls)
Answer : Binomial Distribution…probability of x success in n independent two tailed tests ….
xnx ppxn
xp
)1()(
Binomial Dist.
Mass function for various value of p
n = 15n = 5P = 0.9, 0.5, 0.2
Binomial Distribution
Distribution
Binomial Distribution Mean Variance If Y1, Y2, … Yn are independent bernoulli
RV and Y is bin(n,p) then Y = Y1 + Y2 + …. Yn
If X1, X2… Xm are independent RV and Xi ~ bin(ni,p) then X1 + X2 + … + Xm ~ bin(t1+t2+…….tm, p)
np
)1( pnp
Binomial Distribution The bin(n,p) distribution is symmetric if
and only if p=1/2 X~ bin(n, p) if and only if X ~ bin (n, 1-p) The bin(1,p) and Bernoulli(p)
distributions are same
Geometric Distribution Number of failures before first success in
a sequence of independent Bernoulli trials with probability p of success on each trial…
The probability distribution of the number X of Bernoulli trials needed to get one success…
Geometric Distribution From previous example
Say blue ball = failure Say red ball = success Say we have infinite urns.
Step 1 C = 0 Step 2 Take a new urn Step 3 We pic one ball Step 4 If the ball is red, we are done … Print C
Else If the ball is blue C = C + 1, goto step 2 Now, what is the probability that C will be 5
?? Or 3 ?? Or 0 ??
Geometric Distribution Probability of x failures
= x blue balls followed by 1 red ball So
otherwise ,0
0 if ,)1()(
xppxp
x
x times failure(1-p) to the power xFollowed by 1
success
Geometric Distribution Mean
Variance
MLE :
pp1
2
1pp
1)(1
nXp
Geometric Distribution If X1, X2 … Xs are independent geom(p)
random variables, then X1 + X2 + … + Xs has a negative binomial distribution with parameters s and p
The geometric distribution is the discrete analog of the exponential distribution, in the sense that it is the only discrete distribution with the memoryless property.
The geom(p) distribution is a special case of the negative binomial distribution (with s=1 and the same value for p)
Negative Binomial Distribution Number of failures before the s-th
success in a sequence of independent bernoulli trials with probability p of success on each trial.
Number of good items inspected before encountering the s-th defective item
Number of items in a batch of random size
Number of items demanded from an inventory
Negative Binomial Distribution
Mean :
Variance:
otherwise ,0
0 xif ,)1(1
)(xs pp
xxs
xp
pps )1(
2
)1(pps
Negative Binomial Distribution
Poisson Distribution Number of events that occur in an
interval of time when the events are occuring at a constant rate
Number of items in a batch of random size
Number of items demanded from an inventory
Poisson Distribution
Mean : Variance: MLE :
otherwise ,0
0 if ,!)( xx
exp
x
)(nX
Poisson Distribution If Y1, Y2 …. be a sequence of non
negative IID random variables and let Then the distribution of the Yi‘ If and only if X ~ Poisson(λ)
}1:max{1
i
j
YjiX
}1{exp o
Poisson Distribution If X1, X2, … .Xm are independent Random
variables and Xi ~ Poisson (λi), Then X1+ X2 + X3 …. Xm ~ Poisson (λ1 +λ2 … +λm)