Chap03 - Chapter 3 Discrete Random Variables 3.1 Random...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter 3 Discrete Random Variables 3.1 Random Variables Def 1 A random variable X of a probability space (Ω , A ,P ) is a real-valued function defined on Ω , i.e., X : Ω-→ R Example 1 Tossing two fair dice. Then the sample space will be Ω = { ( a,b ) | a,b ∈ { 1 , 2 , 3 , 4 , 5 , 6 }} For ω = ( a,b ) ∈ Ω , define 1. X ( ω ) = a + b 2. Y ( ω ) = max( a,b ) 3. Z ( ω ) = ( 1 if a = b if a 6 = b Explain the above random variables and discuss their ranges. 1 Example 2 Cut a wire of length l into two segments. The sample space will be Ω = { ( a,b ) | a,b > ,a + b = l } For ω = ( a,b ) ∈ Ω , define 1. X ( ω ) = a 2. Y ( ω ) = min( a,b ) 3. Z ( ω ) = ab Explain the above random variables and discuss their ranges. Remark 1. Although random variables are defined as “functions”, but treating them as “variables” is easier for probability calculation. 2. For example, we will use following notations: P ( X = x ) ≡ P ( { ω | X ( ω ) = x,ω ∈ Ω } ) P ( X > x ) ≡ P ( { ω | X ( ω ) > x,ω ∈ Ω } ) P ( a < X ≤ b ) ≡ P ( { ω | a < X ( ω ) ≤ b,ω ∈ Ω } ) Def 2 A discrete random variable X is a random variable with range being a finite or count- able infinite subset { x 1 ,x 2 ,... } of real numbers R . Example 3 All the random variables defined in Example 1 are discrete, while those in Example 2 are not. 3.2 The Probability Mass Functions (Discrete Density Functions) Def 3 Let X be a discrete random variable. The probability mass function (pmf) of X , denoted by p X ( x ) , is defined as p X ( x ) = P ( X = x ) = X X ( ω )= x P ( ω ) 2 Example 4 Determine the pmf’s for random variables X , Y and Z in Example 1, and plot their corresponding diagrams. sol ) 1. • coef ( z x ) of the following equation gives the number of possible outcomes for X = x . ( z + z 2 + z 3 + z 4 + z 5 + z 6 )( z + z 2 + z 3 + z 4 + z 5 + z 6 ) = z 2 (1 + z + z 2 + z 3 + z 4 + z 5 ) 2 = z 2 (1- z 6 ) 2 (1- z ) 2 = ( z 2- 2 z 8 + z 14 ) ∞ X j =0 ˆ j + 1 j ! z j coef ( z x ) = ˆ x- 1 x- 2 !- 2 ˆ x- 7 x- 8 ! + ˆ x- 13 x- 14 ! =      x- 1 if x = 2 , 3 , 4 , 5 , 6 , 7 13- x if x = 8 , 9 , 10 , 11 , 12 otherwise • Therefore, p X ( x ) =      ( x- 1) / 36 x = 2 , 3 , 4 , 5 , 6 , 7 (13- x ) / 36 x = 8 , 9 , 10 , 11 , 12 otherwise 2. • Let X 1 ,X 2 be the r.v’s representing the outcomes of the 1st and 2nd dice, respec- tively. Then p Y ( y ) = P ( Y = y ) = P [ max ( X 1 ,X 2 ) = y ] • Event analysis: [ max ( X 1 ,X 2 ) = y ] = [1 ≤ x 1 ≤ 6][1 ≤ x 2 ≤ 6][ x 1 = y ][ x 2 ≤ y ] + [1 ≤ x 1 ≤ 6][1 ≤ x 2 ≤ 6][ x 2 = y ][ x 1 < y ] = [1 ≤ x 2 ≤ y ][ x 1 = y ] + [1 ≤ x 1 < y ][ x 2 = y ] • Therefore, p Y ( y ) = P [ max ( X 1 ,X 2 ) = y ] = y X x 2 =1 P ( X 1 = y,X 2 = x 2 ) + y- 1 X x 1 =1 P ( X 1 = x 1 ,X 2 = y ) = y 36 + y- 1 36 = 2 y- 1 36 y = 1 , 2 , 3 , 4 , 5 , 6 3 3. Clearly, p Z ( z ) = P ( Z = z ) =      5 6 z = 0 1 6 z = 1 0 otherwise Remark A pmf must satisfy...
View Full Document

This note was uploaded on 02/13/2012 for the course ECON 101 taught by Professor Teerana during the Spring '11 term at Thammasat University.

Page1 / 30

Chap03 - Chapter 3 Discrete Random Variables 3.1 Random...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online