Download STA 348 Introduction to Stochastic Processes Lecture 2

STA 348
Introduction to
Stochastic Processes
Lecture 2
1
Example

n# points are randomly drawn on a circle.
What is the probability that all points lie in a
semi-circle?
2
Random Variables

A random variable (RV) X is function from
sample space to real numbers X : S  


In other words, for any sample point Ei  S the
random variable assigns a number X  Ei   
E.g. X = # of heads in 2 coin flips
Sample space (S)
E1   H , H  E2   H , T 
E3  T , H  E4  T , T 

X
Values of random variable X
X  E1   2 X  E2   1
X  E3   1 X  E4   0
Describe events using RV’s indirectly, as
 x  all sample points Ei , such that X  Ei   x
3
Discrete RV’s

RV X is discrete, if it assumes finite {x1,...,xn}
or countably infinite {x1,x2,...} # of values.
S
 Discrete RV’s partition
X  x 
2
sample space into countable
number of events


 X  x1

Probability mass function (pmf)
p  x   P  X  x  x, so that: 0  p ( x)  1 &  x p  x   1
Cumulative distribution function (cdf)
F  x   P  X  x  x,
F ( x)  0
 xlim

F ( x) non-decreasing & 
F ( x)  1
 xlim

4
Common Discrete RV’s

Bernoulli: p(1)  p & p(0)  1  p  q, 0  p  1


Outcome is success (X=1) , or failure (X=0)
n x
n x
Binomial: p  x     p 1  p  , 0  x  n
 x

# of successes in n indep. Bernoulli trials
x 1

Geometric: p ( x)  pq , for x  1


# trials till 1st success in series of Bernoulli trials
 x 
Poisson: p ( x)  e , for x  0 with   0
x!

# successes in some interval, with success rate λ
5
Continuous RV’s


Continuous RV has continuous cdf F(x)
Probability density function (pdf) for cont. RV
dF  x 
f  x 
 F '  x  , x where derivative exists
dx

Properties of pdf’s:

f  x   0, x  

f  x  dx  1



P  X  B    f  x  dx, for every B  

B
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Common Continuous RV’s



 b  a 1 , a  x  b
f  x  
otherwise
 0,
Uniform:
Exponential:
Gamma:


 e   x ,
x0
f  x  
otherwise
 0,
 e   x ( x) 1 / ( ),
x0
f  x  
0,
otherwise

( )   x 1e  x dx,
0
& (n)  (n  1)!
2

1
  x    
exp 
 Normal: f  x  
, x  
2
2
2 


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Expectations

Expected value of RV X:




x
Continuous: E  X  
x  p  x



x  f  x  dx
Expected value of function g(·) of RV X:



Discrete: E  X  
Discrete: E  g  X     x g  x   p  x 
Continuous: E  g ( X ) 



g ( x)  f ( x)dx
2
2
Var
(
X
)

E
(
X
)

[
E
(
X
)]
Variance of RV X:
8
Example

For discrete, non-negative RV X, show that:
E ( X )   x1 P  X  x    x 0 P  X  x 
9
Example

For continuous, non-negative RV X, show that:

E ( X )   P  X  x  dx
0
10
Joint Distributions

For 2 discrete RV’s X,Y their joint pmf is
p  x , y   P  X  x, Y  y 


Marginal pmf of X: p X  x    y p  x, y 
For 2 continuous RV’s X,Y their joint pdf is
f  x, y  such that P  X  A, Y  B   
B

A
f ( x, y ) dxdy



Marginal pdf of X: f X  x    f ( x, y )dy
RV’s X,Y are independent iff:
P  X  a, Y  b  
 p  x, y   p X ( x) pY ( y )

P  X  a  P (Y  b)
 f  x, y   f X ( x) fY ( y )
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Expectations & Covariances

Properties of expectations:



Linearity: E  g ( X )  f (Y )  c   E[ g ( X )]  E[ f (Y )]  c
For indep. X, Y: E  g ( X )  f (Y )   E[ g ( X )]  E[ f (Y )]
Covariance: Cov( X , Y )  E ( XY )  E ( X ) E (Y )





Cov( X , X )  Var ( X )
X,Y indep. ⇒ Cov(X,Y) =0, but NOT vice-versa
Cov( a  X  b, c  Y  d )  a  c  Cov( X , Y )
Cov( i X i ,  j Y j )   i  j Cov( X i , Y j )
Var ( i X i )   i Var ( X i )  2 i  j i Cov( X i ,X j )
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