Transcript Chapter 2

Chapter 2
Random Variable & their
Distribution
Illustration
Definition
R.V say X is a function defined over a
sample space S, that associates a real
number, X(e)=x, whith each possible
outcome e in S
Look at example above!
max(1,1) 1, max(2,2) 2, max(3,2) 3, max(4,3) 4
each of theeventsB1 , B2 , B3 , B4 of S contain
thepairs(i, j) other wordX has value
x  1 over B1 , x  2 over B2 , x  3 over B3 , x  4 over B4
Other Example
An experiments involving a sequence of
5 tosses of a coin, the number of Heads
in the sequence is a random variable
Two rolls of a die, v.r :
The sum of the two rolls
The number of sixes in the two rolls
The second roll raised to fifth power
Main Concepts Related to RV
• A RV is a real valued function of the
outcome of the experiment
• A function of R.V defines another R.V
• A R.V can be conditioned on an event
or on another R.V
• There is a notion of independence of a
R.V from an event or from another R.V
Definition
• let us consider functions which take values in the
real numbers.
• In the coin tossing example, our function might
count the number of heads. Call this function R.
We can look at the set
• If we have chosen the set of events to contain all
subsets of , then this set is an event, and we can
ask for the probability of {R=6}
• The precise relation is that if the model is (,F,Pr)
and R:(-,) then for every interval I,
{RI}:={w:R(w)I}F
Definition :
Function which satisfied
Are called (real valued) R.V
Example 1
  1,2,3, F is all subsets of , Pr(A)  number of elementsof A
divided by 3. R(x)  x
R is a R. V since for every interval I the set RI is a
subset of ,
and all subsets of  are in F
Example 2
  1,2,3, F  , ,{1,2}, {3},
Pr(A)  thenumber of elementsof A divided by 3.R( x)  x
R is not a R.V since R=2={2} is not in F
Discrete R.V
R.V si discrete if its range is finite or at
most countably infinite
Definition :
If the set of all possible values of a R.V X
is a countable set, x1, x2 ,, xn atau x1, x2 ,
then X called a discrete R.V
f(x)=P[X=x], x  x1 , x2 ,
called the discrete probability density
function (discrete pdf)
Definition
Example 1
Example 2
• The experiment consist of two
independent tosses of a fair coin, let X
be the number of heads obtained, then
the pdf of X is :
1
 4 , if x  0 or x  2

1
f ( x)  
, if x  1
2

 0, ot herwise

Example 3
If f ( x)  c(2x 1), x  1,2,...,12
Then find c!
EXERCISE
Cummulative Density Function
Definition
Theorem
A function F(x) is a CDF for some R.V X if
and only if it satisfies the following
properties :
1. lim F ( x)  0
x  
2. lim F ( x)  1
x 
3. lim F x  h   F ( x)
h 0
4. a  b implies F (a)  F (b)
Example 1
 0, x  2
0.2,  2 x  0

F ( x)  
 0.7,0  x  2
 1, x  2
Example 2
Suppose that a days production of 850 manufactured
parts contains 50 parts that don’t conform to customer
requirements. Two parts are selected at random,
without replacement, from the batch. Let the random
variable X equal the number of nonconforming parts
in the sample. What the cdf of X?
Exercise