Simple stochastic models 1

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Simple stochastic models 1
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Random variation 1

• Genetic, physiological
• Environmental
• Measurement error
• Random sampling

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Random variation 2

• Random variability as nuisance
• Random variability of primary interest

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Terminology

• Trial
• Sample space
• Event
• Probability
• Random variable

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Discrete random variable X

• Takes values
• x0, x1, x2, .
• with corresponding probabilities P(Xxk)pk
• p0, p1, p2,
• p0 p1 p2 1

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Discrete - integer random variable X

• Takes values
• 0, 1, 2, .
• with corresponding probabilities P(Xk)pk
• p0, p1, p2,
• p0 p1 p2 1

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Mean and variance of discrete random variable
Mean
Variance
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Binomial Distribution (Bernoulli trials)

• n number of trials
• p probability of success
• P(k successes in n trials) nCk pk (1-p)n-k

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Binomial distribution

• X binomial(n,p)
• E(X) n p
• V(X) n p (1 p )

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n 10, p 0.5
0.3
0.25
0.2
P(Xk)
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
k
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Poisson distribution

• X poisson(?)
• k number of events

E(X) ?, V(X) ?
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? 5
0.18
0.16
0.14
0.12
P(Xk)
0.1
0.08
0.06
0.04
0.02
0
0
2
4
6
8
10
12
14
16
18
20
k
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• When number of Bernoulli trials n is large, and
  probability of success p is small, the
  distribution of number of successes becomes
  Poisson.

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Examples

• 30 of women in Germany are smokers. We take a
  random (representative) sample of 20 women. The
  distribution of number of smokers among them is
• X binomial(20,0.3)

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• Probability that an accident leading to injury
  happens in a factory is p0.0001 per day. Number
  of accidents X over a ten year period (n3650) is
  Poisson, XPoisson(?), with
• ? np 0.365

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Generating function of discrete - integer random
variables

• Discrete - integer random variable
• X 0, 1, 2, .
• p0, p1, p2,
• Generating function
• P(s) p0 s p1 s2 p2

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Properties

• P(1) 1
• P(1) E(X)
• P(1) EX(X-1)

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Binomial and Poisson distrib.
Binomial X binomial(n,p), q1-p
Poisson X Poisson(? )
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Continuity theorem for generating functions
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Two dimensional discrete integer random
variables

• Joint probability distribution
• PXY(Xi, Yk) pik
• Two dimensional generating function

Marginal distributions PX(s)PXY(s,1)
PY(s)PXY(1,s)
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Independent discrete integer random variables

• X PX(Xi) pi
• Y PY(Yk) pk
• PXY(Xi, Yk) pi pk

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Sum

• PXY(Xi, Yk) pik
• Z X Y

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Expectation and variance of sums of independent
random variables

• E(XY) E(X) E(Y)
• X, Y - independent
• V(XY) V(X) V(Y)

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Sum

• Z X Y
• PZ(s) PXY(s,s)
• X,Y independent
• PZ(s) PX(s) PY(s)

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Example

• X1 binomial(1,p) (one Bernoulli trial)
• Xn binomial(n,p)
• PX1(s)qsp
• PXn(s)(qsp)n

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Example

• X Poisson(?), YPoisson(?)
• ZXY
• PZ(s) PX(s) PY(s)

Z Poisson(??)
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Can we generalize generating function method to
non integer discrete r.v. ?

• x0, x1,
  x2, .
• P(Xxk)pk p0, p1, p2,